Prove the following trigonometric identities.
Question:

Prove the following trigonometric identities.

$\frac{1}{\sec A-1}+\frac{1}{\sec A+1}=2 \operatorname{cosec} A \cot A$

Solution:

We need to prove $\frac{1}{\sec A-1}+\frac{1}{\sec A+1}=2 \operatorname{cosec} A \cot A$

Solving the L.H.S, we get

$\frac{1}{\sec A-1}+\frac{1}{\sec A+1}=\frac{\sec A+1+\sec A-1}{(\sec A-1)(\sec A+1)}$

$=\frac{2 \sec A}{\sec ^{2} A-1}$

Further using the property $1+\tan ^{2} \theta=\sec ^{2} \theta$, we get

So,

$\frac{2 \sec A}{\sec ^{2} A-1}=\frac{2 \sec A}{\tan ^{2} A}$

$=\frac{2\left(\frac{1}{\cos A}\right)}{\frac{\sin ^{2} A}{\cos ^{2} A}}$

$=2 \frac{1}{\cos A} \times \frac{\cos ^{2} A}{\sin ^{2} A}$

$=2\left(\frac{\cos A}{\sin A}\right) \times \frac{1}{\sin A}$

$=2 \operatorname{cosec} A \cot A$

Hence proved.

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