Prove that : (iii) $\sin \left(28^{\circ}+A\right)=\cos \left(62^{\circ}-A\right)$ (iv) $1 /\left(1+\cos \left(90^{\circ}-A\right)\right)+1 /\left(1-\cos \left(90^{\circ}-A\right)\right)=2 \operatorname{cosec}^{2}\left(90^{\circ}-A\right)$
(iii) $\sin \left(28^{\circ}+\mathrm{A}\right)=\sin \left[90^{\circ}-\left(62^{\circ}-\mathrm{A}\right)\right]=\cos \left(62^{\circ}-\mathrm{A}\right)$ (iv)
Prove that: (i) $\tan \left(55^{\circ}+x\right)=\cot \left(35^{0}-x\right)$ (ii) $\sec \left(70^{\circ}-\theta\right)=\operatorname{cosec}\left(20^{\circ}+\theta\right)$
(i) $\tan \left(55^{\circ}+x\right)=\tan \left[90^{\circ}-\left(35^{\circ}-x\right)\right]=\cot \left(35^{\circ}-x\right)$ (ii) $\sec \left(70^{\circ}-\theta\right)=\sec...
Evaluate: (vii) $3cos\mathrm{} 80^{\circ} \operatorname{cosec} 10^{\circ}+2 \cos 59^{\circ} \operatorname{cosec} 31^{\circ}$ (viii) $\frac{\cos 75^{\circ}}{\sin 15^{\circ}}+\frac{\sin 12^{\circ}}{\cos 78^{\circ}}-\frac{\cos 18^{\circ}}{\sin 72^{\circ}}$
(vii) $3 \cos 80^{\circ} \operatorname{cosec} 10^{\circ}+2 \cos 59^{\circ} \operatorname{cosec} 31^{\circ}$ $=3 \cos (90-10)^{0} \operatorname{cosec} 10^{\circ}+2 \cos (90-31)^{0}...
Evaluate : (v) $\sin 27^{\circ} \sin 63^{\circ}-\cos 63^{\circ} \cos 27^{\circ}$ (vi) $\frac{3 \sin 72^{\circ}}{\cos 18^{\circ}}-\frac{\sec 32^{\circ}}{\operatorname{cosec} 58^{\circ}}$
(v) $\sin 27^{\circ} \sin 63^{\circ}-\cos 63^{\circ} \cos 27^{\circ}$ $=\sin (90-63)^{0} \sin 63^{\circ}-\cos 63^{\circ} \cos (90-63)^{\circ}$ $=\cos 63^{\circ} \sin 63^{\circ}-\cos 63^{\circ} \sin...
Evaluate : (iii) $\frac{5 \sin 66^{\circ}}{\cos 24^{\circ}}-\frac{2 \cot 85^{\circ}}{\tan 5^{\circ}}$ (iv) $\cos 40^{\circ} \operatorname{cosec} 50^{\circ}+\sin 50^{\circ} \sec 40^{\circ}$
(iii) (iv) $\cos 40^{\circ} \operatorname{cosec} 50^{\circ}+\sin 50^{\circ}$ sec $40^{\circ}$ $=\cos (90-50)^{0} \operatorname{cosec} 50^{\circ}+\sin (90-50)^{0} \sec 40^{\circ}$ $=\sin 50^{\circ}...
Evaluate:(i) $2\left(\frac{\tan 35^{\circ}}{\cos^{\circ} 55^{\circ}}\right)^{2}+\left(\frac{\cos+55^{\circ}}{\tan 35^{\circ}}\right)^{2}-3\left(\frac{\sec 40^{\circ}}{cosec 50^{\circ}}\right)$ (ii) $\sec 26^{\circ} \sin 64^{\circ}+\frac{\operatorname{co} \sec 33^{\circ}}{\sec 57^{\circ}}$
(i) (ii) 1+1=2
(i) If $2 \sin \mathrm{A}-1=0$, show that: $\sin 3 A=3 \sin A-4 \sin ^{3} A$ (ii) If $4 \cos ^{2} \mathrm{~A}-3=0$, show that: $\cos 3 A=4 \cos ^{2} A-3 \cos A$
(i) Since, $2 \sin \mathrm{A}-1=0$ Therefore, $\sin A=1 / 2$ since, $\sin 30^{\circ}=1 / 2$ Hence, $\mathrm{A}=30^{\circ}$ LHS = $\sin 3 A=\sin 3\left(30^{\circ}\right)=\sin 30^{\circ}=1$...
If tan A = n tan B and sin A = m sin B, prove that: cos^2 A = m^2 – 1/ n^2 – 1
$\tan A=n \tan B$ $n=\tan A / \tan B$ And, $\sin A=m \sin B$ $\mathrm{m}=\sin \mathrm{A} / \sin \mathrm{B}$ Substituting RHS in $\mathrm{m}$ and $\mathrm{n}$ $m^{2}-1 / n^{2}-1$...
If $\sec \mathbf{A}+\tan \mathbf{A}=\mathbf{p}$, show that: $\sin A=\left(p^{2}-1\right) /\left(p^{2}+1\right)$
RHS, $\left(p^{2}-1\right) /\left(p^{2}+1\right)$ $=\frac{(\sec A+\tan A)^{2}-1}{(\sec A+\tan A)^{2}+1}$ $=\frac{\sec ^{2} A+\tan ^{2} A+2 \tan A \sec A-1}{\sec ^{2} A+\tan ^{2} A+2 \tan A \sec...
If $x=a \cos \theta$ and $y=b \cot \theta$, show that: $a^{2} / x^{2}-b^{2} / y^{2}=1$
LHS = $a^{2} / x^{2}-b^{2} / y^{2}$ $=\frac{a^{2}}{a^{2} \cos ^{2} \theta}-\frac{b^{2}}{b^{2} \cot ^{2} \theta}$ $=\frac{1}{\cos ^{2} \theta}-\frac{\sin ^{2} \theta}{\cos ^{2} \theta}$...
If sin A + cos A = p and sec A + cosec A = q, then prove that: q(p^2 – 1) = 2p
LHS = $q\left(p^{2}-1\right)=(\sec A+\operatorname{cosec} A)\left[(\sin A+\cos A)^{2}-1\right]$ $=(\sec A+\operatorname{cosec} A)\left[\sin ^{2} A+\cos ^{2} A+2 \sin A \cos A-1\right]$ $=(\sec...
Prove: (xv) $\sec ^{4} A\left(1-\sin ^{4} A\right)-2 \tan ^{2} A=1$ (xvi) $\cos e c^{4} A\left(1-\cos ^{4} A\right)-2 \cot ^{2} A=1$
(xv) LHS = $\sec ^{4} \mathrm{~A}\left(1-\sin ^{4} \mathrm{~A}\right)-2 \tan ^{2} \mathrm{~A}$ $=\sec ^{4} \mathrm{~A}\left(1-\sin ^{2} \mathrm{~A}\right)\left(1+\sin ^{2} \mathrm{~A}\right)-2 \tan...
Prove : (xvii) $(1+\tan A+\sec A)(1+\cot A-\operatorname{cosec} A)=2$
$(1+\tan A+\sec A)(1+\cot A-\operatorname{cosec} A)$ $=1+\cot A-\operatorname{cosec} A+\tan A+1-\sec A+\sec A+\operatorname{cosec} A-\operatorname{cosec} A \sec A$ $=2+\cos \mathrm{A} / \sin...
Prove : (xiii) $\cot ^{2} A\left(\frac{\sec A-1}{1+\sin A}\right)+\sec ^{2} A\left(\frac{\sin A-1}{1+\sec A}\right)=0$ (xiv) $\frac{\left(1-2 \sin ^{2} A\right)^{2}}{\cos ^{4} A-\sin ^{4} A}=2 \cos ^{2} A-1$
LHS = = RHS (xiv) LHS = = RHS
Prove : (xi) $\frac{1+(\sec A-\tan A)^{2}}{\operatorname{cosec} A(\sec A-\tan A)}=2 \tan A$ (xii) $\frac{(\operatorname{cosec} A-\cot A)^{2}+1}{\sec A(\operatorname{cosec} A-\cot A)}=2 \cot A$
LHS = = RHS (xii) LHS = = RHS
Prove (ix) $\sqrt{\frac{1+\sin A}{1-\sin A}}=\frac{\cos A}{1-\sin A}$ (x) $\sqrt{\frac{1-\cos A}{1+\cos A}}=\frac{\sin A}{1+\cos A}$
LHS= = RHS (x) LHS =
Prove : (vii) $\frac{\sin A}{1-\cos A}-\infty \cos A=\infty$ (viii) $\frac{\sin A-\cos A+1}{\sin A+\cos A-1}=\frac{\cos A}{1-\sin A}$
(vii) LHS =$=(\sin A /(1-\cos A))-\cot A$ Since, $\cot A=\cos A / \sin A$ $=\left(\sin ^{2} A-\cos A+\cos ^{2} A\right) /(1-\cos A) \sin A$ $=(1-\cos A) /(1-\cos A) \sin A$ $=1 / \sin \mathrm{A}$...
Prove: (v) $\frac{\cot A}{1-\tan A}+\frac{\tan A}{1-\cot A}=1+\tan A+\cot A$ (vi) $\frac{\cos A}{1+\sin A}+\tan A=\sec A$
(v) LHS= cot A/ (1 – tan A) + tan A/ (1 – cot A) = RHS (vi) LHS= cos A/ (1 + sin A) + tan A = RHS
Prove: (iii) $1-\frac{\sin ^{2} A}{1+\cos A}=\cos A$ (iv) $\frac{1-\cos A}{\sin A}+\frac{\sin A}{1-\cos A}=2 \operatorname{cosec} A$
(iii) LHS= 1 – sin2 A/ (1 + cos A) = RHS (iv) LHS= (1 – cos A)/ sin A + sin A/ (1 – cos A) = RHS
Use tables to find sine of: (i) 21° (ii) 34° 42′
(i) Taking LHS, $1 /(\cos A+\sin A)+1 /(\cos A-\sin A)$ (ii) Taking LHS, $\operatorname{cosec} A-\cot A$ $=\frac{1}{\sin A}-\frac{\cos A}{\sin A}$ $=\frac{1-\cos A}{\sin A}$ $=\frac{1-\cos A}{\sin...
Use trigonometrical tables to find tangent of: (iii) 17° 27′
(iii) $\tan 17^{\circ} 27^{\prime}=\tan \left(17^{\circ} 24^{\prime}+3^{\prime}\right)=0.3134+0.0010=0.3144$
Use trigonometrical tables to find tangent of: (i) 37° (ii) 42° 18′
(i) $\tan 37^{\circ}=0.7536$ (ii) $\tan 42^{\circ} 18^{\prime}=0.9099$
Use tables to find cosine of: (v) 9° 23’ + 15° 54’
$(\mathrm{v}) \cos \left(9^{\circ} 23^{\circ}+15^{\circ} 54^{\circ}\right)=\cos 24^{\circ} 77^{\circ}=\cos 25^{\circ} 17^{\circ}=\cos \left(25^{\circ}...
Use tables to find cosine of: (iii) 26° 32’ (iv) 65° 41’
(iii) $\cos 26^{\circ} 32^{\prime}=\cos \left(26^{\circ} 30^{\prime}+2^{\prime}\right)=0.8949-0.0003=0.8946$ (iv) $\cos 65^{\circ} 41^{\prime}=\cos \left(65^{\circ}...
Use tables to find cosine of: (i) 2° 4’ (ii) 8° 12’
(i) $\cos 2^{\circ} 4^{\prime}=0.9994-0.0001=0.9993$ (ii) $\cos 8^{\circ} 12^{\prime}=\cos 0.9898$
Use tables to find sine of: (v) 10° 20′ + 20° 45′
(v) $\sin \left(10^{\circ} 20^{\prime}+20^{\circ} 45^{\prime}\right)=\sin 30^{\circ} 65^{\prime}=\sin 31^{\circ} 5^{\prime}=0.5150+0.0012=0.5162$
Use tables to find sine of: (iii) 47° 32′ (iv) 62° 57′
(iii) $\sin 47^{\circ} 32^{\prime}=\sin \left(47^{\circ} 30^{\prime}+2^{\prime}\right)=0.7373+0.0004=0.7377$ (iv) $\sin 62^{\circ} 57^{\prime}=\sin \left(62^{\circ}...
Use tables to find sine of: (i) 21° (ii) 34° 42′
(i) $\sin 21^{\circ}=0.3584$ (ii) $\sin 34^{\circ} 42^{\prime}=0.5693$
A triangle ABC is right angled at B; find the value of (sec A. cosec C – tan A. cot C)/ sin B
Since, ABC is a right angled triangle right angled at B Therefore, \[\begin{array}{*{35}{l}} A\text{ }+\text{ }C\text{ }=\text{ }{{90}^{o}} \\ \left( sec\text{ }A.\text{ }cosec\text{ }C\text{...
Evaluate: (ix) $14 \sin 30^{\circ}+6 \cos 60^{\circ}-5 \tan 45^{\circ}$
(ix) \[14\text{ }sin\text{ }{{30}^{o}}~+\text{ }6\text{ }cos\text{ }{{60}^{o}}~-\text{ }5\text{ }tan\text{ }{{45}^{o}}\] \[=\text{ }14\text{ }\left( 1/2 \right)\text{ }+\text{ }6\text{ }\left( 1/2...
Evaluate: (vii) $\frac{\operatorname{cot}^{2} 41^{\circ}}{\tan ^{2} 49^{\circ}}-2 \frac{\sin ^{2} 75^{\circ}}{\cos ^{2} 15^{\circ}}$ (viii) $\frac{\cos 70^{\circ}}{\sin 20^{\circ}}+\frac{\cos 59^{\circ}}{\sin 31^{\circ}}-8 \sin ^{2} 30^{\circ}$
(vii) = 1 – 2 = -1 (viii)
Evaluate: (v) $\operatorname{cosec}\left(65^{\circ}+A\right)-\sec \left(25^{\circ}-A\right)$ (vi) $2 \frac{\tan 57^{\circ}}{\cot 33^{\circ}}-\frac{\infty+70^{\circ}}{\tan 20^{\circ}}-\sqrt{2} \cos 45^{\circ}$
\[\begin{array}{*{35}{l}} v)\text{ }cosec\text{ }\left( {{65}^{o}}~+\text{ }A \right)\text{ }-\text{ }sec\text{ }\left( {{25}^{o}}~-\text{ }A \right) \\ =\text{ }cosec\text{ }\left[...
Evaluate: (iii) $\frac{\sin 80^{\circ}}{\cos 10^{\circ}}+\sin 59^{\circ} \sec 31^{\circ}$ (iv) $\tan \left(55^{\circ}-A\right)-\cot \left(35^{\circ}+A\right)$
\[\begin{array}{*{35}{l}} \left( iii \right)\text{ }sin\text{ }{{80}^{o}}/\text{ }cos\text{ }{{10}^{o}}~+\text{ }sin\text{ }{{59}^{o}}~sec\text{ }{{31}^{o}} \\ =\text{ }sin\text{ }{{\left( 90\text{...
Evaluate: (i) $3 \frac{\sin 72^{\circ}}{\cos 18^{\circ}}-\frac{\sec 32^{\circ}}{\operatorname{cosec} 58^{\circ}}$ (ii) $3 \cos 80^{\circ} \operatorname{cosec} 10^{\circ}+2 \cos 59^{\circ}$ cosec $31^{\circ}$
(i) (ii) \[\begin{array}{*{35}{l}} 3\text{ }cos\text{ }{{80}^{o}}~cosec\text{ }{{10}^{o}}~+\text{ }2\text{ }cos\text{ }{{59}^{o}}~cosec\text{ }{{31}^{o}} \\ =\text{ }3\text{ }cos\text{ }{{\left(...
For triangle ABC, show that: (i) sin (A + B)/ 2 = cos C/2 (ii) tan (B + C)/ 2 = cot A/2
$\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180^{\circ} \quad$ [Angle sum property of a triangle] $(\angle \mathrm{A}+\angle \mathrm{B}) / 2=90^{\circ}-\angle \mathrm{C} / 2$ $\sin ((A+B)...
Show that: (i) $\frac{\sin A}{\sin \left(90^{\circ}-A\right)}+\frac{\cos A}{\cos \left(90^{\circ}-A\right)}=\sec A \operatorname{cosec} A$ (ii) $\sin A \cos A-\frac{\sin A \cos \left(90^{\circ}-A\right) \cos A}{\operatorname{sed}\left(90^{\circ}-A\right)}-\frac{\cos A \sin \left(90^{\circ}-A\right) \sin A}{\operatorname{cosec}\left(90^{\circ}-A\right)}=0$
\[\begin{array}{*{35}{l}} =\text{ }sin\text{ }A\text{ }cos\text{ }A\text{ }\text{ }si{{n}^{3}}~A\text{ }cos\text{ }A\text{ }\text{ }co{{s}^{3}}~A\text{ }sin\text{ }A \\ =\text{ }sin\text{ }A\text{...
Express each of the following in terms of angles between 0°and 45°: (iii) cos 74°+ sec 67°
\[\begin{array}{*{35}{l}} \left( iii \right)\text{ }cos\text{ }74{}^\circ +\text{ }sec\text{ }67{}^\circ \\ =\text{ }cos~\left( 90\text{ }-\text{ }16 \right){}^\circ +\text{ }sec\text{ }\left(...
Express each of the following in terms of angles between 0°and 45°: (i) sin 59°+ tan 63° (ii) cosec 68°+ cot 72°
\[\begin{array}{*{35}{l}} \left( i \right)\text{ }sin\text{ }59{}^\circ +\text{ }tan\text{ }63{}^\circ \\ =\text{ }sin\text{ }\left( 90\text{ }-\text{ }31 \right){}^\circ +\text{ }tan\text{...
Show that: (iii) sin26/ sec64 + cos 26/ cosec 64 = 1
Show that: (i) tan10 tan15 tan75 tan80 = 1 (ii) sin42 sec48 + cos42 cosec48 = 2
(i) \[sin\text{ }{{42}^{o}}~sec\text{ }{{48}^{o}}~+\text{ }cos\text{ }{{42}^{o}}~cosec\text{ }{{48}^{o}}\] \[~=\text{ }sin\text{ }{{42}^{o}}~sec\text{ }({{90}^{o}}~-\text{ }{{42}^{o}})\text{...
If $m=a \sec A+b \tan A$ and $n=a \tan A+b \sec A$, prove that $m^{2}-n^{2}=a^{2}-b^{2}$
Taking LHS, $\mathrm{m}^{2}-\mathrm{n}^{2}$ $=(a \sec A+b \tan A)^{2}-(a \tan A+b \sec A)^{2}$ $=a^{2} \sec ^{2} A+b^{2} \tan ^{2} A+2 a b \sec A \tan A-a^{2} \tan ^{2} A-b^{2} \sec ^{2} A-2 a b...
Prove: (ix) $\frac{1}{\cos A+\sin A-1}+\frac{1}{\cos A+\sin A+1}=\operatorname{cosec} A+\sec A$
Prove: (vii) $(\operatorname{cosec} A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$ (viii) $(1+\tan A \tan B)^{2}+(\tan A-\tan B)^{2}=\sec ^{2} A \sec ^{2} B$
(vii) (viii)
Prove: (v) $2 \sin ^{2} A+\cos ^{4} A=1+\sin ^{4} A$ (vi) $\frac{\sin A-\sin B}{\cos A+\cos B}+\frac{\cos A-\cos B}{\sin A+\sin B}=0$
(v) \[\begin{array}{*{35}{l}} 2\text{ }si{{n}^{2}}~A\text{ }+\text{ }co{{s}^{2}}~A \\ =\text{ }2\text{ }si{{n}^{2}}~A\text{ }+\text{ }{{\left( 1\text{ }-\text{ }si{{n}^{2}}~A \right)}^{2}} \\...
Prove: (iii) $\frac{\tan A}{1-\cot A}+\frac{\infty \tan A}{1-\tan A}=\sec A \operatorname{cosec} A+1$ (iv) $\left(\tan A+\frac{1}{\cos A}\right)^{2}+\left(\tan A-\frac{1}{\cos A}\right)^{2}=2\left(\frac{1+\sin ^{2} A}{1-\sin ^{2} A}\right)$
(iii) (iv)
Prove: (i). cosA/1-tanA+ sinA/1-cotA = sinA + cosA (ii) $\frac{\cos ^{3} A+\sin ^{3} A}{\cos A+\sin A}+\frac{\cos ^{3} A-\sin ^{3} A}{\cos A-\sin A}=2$
(i) (ii)
Prove: 1/ 1 – sin A + 1/ 1 + sin A = 2 sec^2 A
Prove: 1/ 1 + cos A + 1/ 1 – cos A = 2 cosec^2A
Prove: \[~\mathbf{se}{{\mathbf{c}}^{\mathbf{2}}}~\mathbf{A}.\text{ }\mathbf{cose}{{\mathbf{c}}^{\mathbf{2}}}~\mathbf{A}\text{ }=\text{ }\mathbf{ta}{{\mathbf{n}}^{\mathbf{2}}}~\mathbf{A}\text{ }+\text{ }\mathbf{co}{{\mathbf{t}}^{\mathbf{2}}}~\mathbf{A}\text{ }+\text{ }\mathbf{2}\]
\[\begin{array}{*{35}{l}} RHS\text{ }=\text{ }ta{{n}^{2}}~A\text{ }+\text{ }co{{t}^{2}}~A\text{ }+\text{ }2\text{ }=\text{ }ta{{n}^{2}}~A\text{ }+\text{ }co{{t}^{2}}~A\text{ }+\text{ }2\text{...
Prove: \[{{\left( \mathbf{sin}\text{ }\mathbf{A}\text{ }+\text{ }\mathbf{cosec}\text{ }\mathbf{A} \right)}^{\mathbf{2}}}~+\text{ }{{\left( \mathbf{cos}\text{ }\mathbf{A}\text{ }+\text{ }\mathbf{sec}\text{ }\mathbf{A} \right)}^{\mathbf{2}}}~=\text{ }\mathbf{7}\text{ }+\text{ }\mathbf{ta}{{\mathbf{n}}^{\mathbf{2}}}~\mathbf{A}\text{ }+\text{ }\mathbf{co}{{\mathbf{t}}^{\mathbf{2}}}~\mathbf{A}\]
\[\begin{array}{*{35}{l}} LHS, \\ {{\left( sin\text{ }A\text{ }+\text{ }cosec\text{ }A \right)}^{2}}~+\text{ }{{\left( cos\text{ }A\text{ }+\text{ }sec\text{ }A \right)}^{2}} \\ =\text{...
Prove: sec A – tan A/ sec A + tan A = 1 – 2 secA tanA + 2 tan^2 A
= 1 + tan2 A + tan2 A – 2 sec A tan A = 1 – 2 sec A tan A + 2 tan2 A = RHS
Prove: cosec A + cot A = 1/ cosec A – cot A
cosec A + cot A
Prove: 1/ sec A + tan A = sec A – tan A
Prove: (cosec A – sin A)(sec A – cos A)(tan A + cot A) = 1
(cosec A – sin A)(sec A – cos A)(tan A + cot A)
Prove: \[~{{\left( \mathbf{cos}\text{ }\mathbf{A}\text{ }+\text{ }\mathbf{sin}\text{ }\mathbf{A} \right)}^{\mathbf{2}}}~+\text{ }{{\left( \mathbf{cosA}\text{ }-\text{ }\mathbf{sin}\text{ }\mathbf{A} \right)}^{\mathbf{2}}}~=\text{ }\mathbf{2}\]
\[\begin{array}{*{35}{l}} {{\left( cos\text{ }A\text{ }+\text{ }sin\text{ }A \right)}^{2}}~+\text{ }{{\left( cosA\text{ }-\text{ }sin\text{ }A \right)}^{2}} \\ =\text{ }cos2\text{ }A\text{ }+\text{...
Prove: \[\left( \mathbf{sec}\text{ }\mathbf{A}\text{ }-\text{ }\mathbf{cos}\text{ }\mathbf{A} \right)\left( \mathbf{sec}\text{ }\mathbf{A}\text{ }+\text{ }\mathbf{cos}\text{ }\mathbf{A} \right)\text{ }=\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}~\mathbf{A}\text{ }+\text{ }\mathbf{ta}{{\mathbf{n}}^{\mathbf{2}}}~\mathbf{A}\]
\[\begin{array}{*{35}{l}} \left( sec\text{ }A\text{ }-\text{ }cos\text{ }A \right)\left( sec\text{ }A\text{ }+\text{ }cos\text{ }A \right) \\ =\text{ }\left( se{{c}^{2}}~A\text{ }-\text{...
Prove: \[\left( \mathbf{cosec}\text{ }\mathbf{A}\text{ }+\text{ }\mathbf{sin}\text{ }\mathbf{A} \right)\text{ }\left( \mathbf{cosec}\text{ }\mathbf{A}\text{ }-\text{ }\mathbf{sin}\text{ }\mathbf{A} \right)\text{ }=\text{ }\mathbf{co}{{\mathbf{t}}^{\mathbf{2}}}~\mathbf{A}\text{ }+\text{ }\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}~\mathbf{A}\]
\[\begin{array}{*{35}{l}} \left( cosec\text{ }A\text{ }+\text{ }sin\text{ }A \right)\text{ }\left( cosec\text{ }A\text{ }-\text{ }sin\text{ }A \right) \\ =\text{ }cose{{c}^{2}}~A\text{ }-\text{...
Prove: \[\mathbf{co}{{\mathbf{t}}^{\mathbf{2}}}~\mathbf{A}\text{ }-\text{ }\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}~\mathbf{A}\text{ }=\text{ }\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}\mathbf{A}.\text{ }\mathbf{co}{{\mathbf{t}}^{\mathbf{2}}}\mathbf{A}\]
cot2 A – cos2 A
Prove: \[\mathbf{ta}{{\mathbf{n}}^{\mathbf{2}}}~\mathbf{A}\text{ }-\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}~\mathbf{A}\text{ }=\text{ }\mathbf{ta}{{\mathbf{n}}^{\mathbf{2}}}~\mathbf{A}.\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}~\mathbf{A}\]
tan2 A – sin2 A
Prove: \[\left( \mathbf{1}\text{ }+\text{ }\mathbf{ta}{{\mathbf{n}}^{\mathbf{2}}}~\mathbf{A} \right)\text{ }\mathbf{cot}\text{ }\mathbf{A}/\text{ }\mathbf{cose}{{\mathbf{c}}^{\mathbf{2}~}}\mathbf{A}\text{ }=\text{ }\mathbf{tan}\text{ }\mathbf{A}\]
Prove \[\mathbf{se}{{\mathbf{c}}^{\mathbf{2}}}~\mathbf{A}\text{ }+\text{ }\mathbf{cose}{{\mathbf{c}}^{\mathbf{2}}}~\mathbf{A}\text{ }=\text{ }\mathbf{se}{{\mathbf{c}}^{\mathbf{2}}}~\mathbf{A}\text{ }.\text{ }\mathbf{cose}{{\mathbf{c}}^{\mathbf{2}}}~\mathbf{A}\]
Prove: cosec A (1 + cos A) (cosec A – cot A) = 1
Prove: sec A (1 – sin A) (sec A + tan A) = 1
sec A (1 – sin A) (sec A + tan A)
Prove: cosec^4 A – cosec^2 A = cot^4 A + cot^2 A
\[\begin{array}{*{35}{l}} cose{{c}^{4}}~A\text{ }-\text{ }cose{{c}^{2}}~A \\ =\text{ }cose{{c}^{2}}~A\left( cose{{c}^{2}}~A\text{ }-\text{ }1 \right) \\ =\text{ }\left( 1\text{ }+\text{...
Prove: (1 – tan A)^2 + (1 + tan A)^2 = 2sec^2 A
\[\begin{array}{*{35}{l}} {} \\ {{\left( 1\text{ }-\text{ }tan\text{ }A \right)}^{2}}~+\text{ }{{\left( 1\text{ }+\text{ }tan\text{ }A \right)}^{2}} \\ =\text{ }\left( 1\text{ }+\text{...
Prove the following : sin^4 A – cos^4 A = 2 sin^2 A – 1
\[\begin{array}{*{35}{l}} \mathbf{L}.\mathbf{H}.\mathbf{S}, \\ si{{n}^{4}}~A\text{ }-\text{ }co{{s}^{4}}~A \\ =\text{ }{{\left( si{{n}^{2~}}A \right)}^{2}}~-\text{ }{{\left( co{{s}^{2}}~A...