Exercise 21A
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...