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  • Prove the following identities, where the angles involved are acute angles for which the expressions are defined. 1 + sec A/sec A=sin^2 A/1-cos A

Q.5.    Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

          (iv)\frac{1+\sec A}{\sec A}=\frac{\sin ^{2}A}{1-\cos A}

          [Hint : Simplify LHS and RHS separately]
 

Answers (1)

best_answer

We need to prove-
\frac{1+\sec A}{\sec A}=\frac{\sin ^{2}A}{1-\cos A}

taking LHS;

\\\Rightarrow \frac{1+\sec A}{\sec A}\\ \Rightarrow (1+\frac{1}{\cos A})/\sec A\\ \Rightarrow 1+\cos A

Taking RHS;
We know that identity 1-\cos^2\theta = \sin^2\theta

\\\Rightarrow \frac{\sin ^{2}A}{1-\cos A}\\ \Rightarrow \frac{1-\cos^2 A}{1-\cos A}\\ \Rightarrow \frac{(1-\cos A)(1+\cos A)}{(1-\cos A)}\\ \Rightarrow 1+\cos A

LHS = RHS 

Hence proved.

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manish

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