Question:
$\cos ^{4} A-\sin ^{4} A$ is equal to
(a) $2 \cos ^{2} A+1$
(b) $2 \cos ^{2} A-1$
(c) $2 \sin ^{2} A-1$
(d) $2 \sin ^{2} A+1$
Solution:
The given expression is $\cos ^{4} A-\sin ^{4} A$.
Factorising the given expression, we have
$\cos ^{4} A-\sin ^{4} A$
$=\left(\cos ^{2} A+\sin ^{2} A\right) \times\left(\cos ^{2} A-\sin ^{2} A\right)$
$=1 \times\left(\cos ^{2} A-\sin ^{2} A\right)$
$=\cos ^{2} A-\sin ^{2} A$
$=\cos ^{2} A-\left(1-\cos ^{2} A\right)$
$=\cos ^{2} A-1+\cos ^{2} A$
$=2 \cos ^{2} A-1$
Therefore, the correct option is (b).
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.