Login/Sign Up
Prove the following identities: \[\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}~\left( \mathbf{\pi }/\mathbf{8}\text{ }+\text{ }\mathbf{x}/\mathbf{2} \right)\text{ }\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}~\left( \mathbf{\pi }/\mathbf{8}\text{ }\text{ }\mathbf{x}/\mathbf{2} \right)\text{ }=\text{ }\mathbf{1}/\surd \mathbf{2}\text{ }\mathbf{sin}\text{ }\mathbf{x}\]
Prove the following identities: \[\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}~\left( \mathbf{\pi }/\mathbf{8}\text{ }+\text{ }\mathbf{x}/\mathbf{2} \right)\text{ }\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}~\left( \mathbf{\pi }/\mathbf{8}\text{ }\text{ }\mathbf{x}/\mathbf{2} \right)\text{ }=\text{ }\mathbf{1}/\surd \mathbf{2}\text{ }\mathbf{sin}\text{ }\mathbf{x}\]
CBSE | Class 11 | Exercise 9.1 | Maths | RD Sharma | Trigonometric Functions at Multiples and Submultiples of an Angle
Prove the following identities: \[\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}~\left( \mathbf{\pi }/\mathbf{8}\text{ }+\text{ }\mathbf{x}/\mathbf{2} \right)\text{ }\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}~\left( \mathbf{\pi }/\mathbf{8}\text{ }\text{ }\mathbf{x}/\mathbf{2} \right)\text{ }=\text{ }\mathbf{1}/\surd \mathbf{2}\text{ }\mathbf{sin}\text{ }\mathbf{x}\]

 

Take LHS:

\[si{{n}^{2}}~\left( \pi /8\text{ }+\text{ }x/2 \right)\text{ }\text{ }si{{n}^{2}}~\left( \pi /8\text{ }\text{ }x/2 \right)\]

From formula, \[\text{ }si{{n}^{2}}~A\text{ }\text{ }si{{n}^{2}}~B\text{ }=\text{ }sin\text{ }\left( A+B \right)\text{ }sin\text{ }\left( A-B \right)\]

SO,

\[si{{n}^{2}}~\left( \pi /8\text{ }+\text{ }x/2 \right)\text{ }\text{ }si{{n}^{2}}~\left( \pi /8\text{ }\text{ }x/2 \right)\text{ }=\text{ }sin\text{ }\left( \pi /8\text{ }+\text{ }x/2\text{ }+\text{ }\pi /8\text{ }\text{ }x/2 \right)\text{ }sin\text{ }\left( \pi /8\text{ }+\text{ }x/2\text{ }\text{ }\left( \pi /8\text{ }\text{ }x/2 \right) \right)\]

\[=\text{ }sin\text{ }\left( \pi /8\text{ }+\text{ }\pi /8 \right)\text{ }sin\text{ }\left( \pi /8\text{ }+\text{ }x/2\text{ }\text{ }\pi /8\text{ }+\text{ }x/2 \right)\]

\[=\text{ }sin\text{ }\pi /4\text{ }sin\text{ }x\]

\[=\text{ }1/\surd 2\text{ }sin\text{ }x\text{ }\left[ since,\text{ }since\text{ }\pi /4\text{ }=\text{ }1/\surd 2 \right]\]

= RHS

Hence proved.

i

More Material