# Prove the following trigonometric identities.

Question:

Prove the following trigonometric identities.

$\sec ^{4} A\left(1-\sin ^{4} A\right)-2 \tan ^{2} A=1$

Solution:

We have to prove $\sec ^{4} A\left(1-\sin ^{4} A\right)-2 \tan ^{2} A=1$

We know that, $\sin ^{2} A+\cos ^{2} A=1$

So,

$\sec ^{4} A\left(1-\sin ^{4} A\right)-2 \tan ^{2} A=\frac{1}{\cos ^{4} A}\left(1-\sin ^{4} A\right)-2 \frac{\sin ^{2} A}{\cos ^{2} A}$

$=\left(\frac{1}{\cos ^{4} A}-\frac{\sin ^{4} A}{\cos ^{4} A}\right)-2 \frac{\sin ^{2} A}{\cos ^{2} A}$

$=\left(\frac{1-\sin ^{4} A}{\cos ^{4} A}\right)-2 \frac{\sin ^{2} A}{\cos ^{2} A}$

$=\frac{\left(1-\sin ^{2} A\right)\left(1+\sin ^{2} A\right)}{\cos ^{4} A}-2 \frac{\sin ^{2} A}{\cos ^{2} A}$

$=\frac{\cos ^{2} A\left(1+\sin ^{2} A\right)}{\cos ^{4} A}-2 \frac{\sin ^{2} A}{\cos ^{2} A}$

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

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

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

$=\frac{\cos ^{2} A}{\cos ^{2} A}$

$=1$

Hence proved.