# cos4 A − sin4 A is equal to

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).