Question

Solve:

${\cos }^{4}x-{\sin }^{4}x=$

• A. $2{\sin }^{2}x-1$
• B. $2{\cos }^{2}x-1$
• C. $1+2{\sin }^{2}x$
• D. $1-2{\cos }^{2}x$

Answer 1

The correct option is B $2{\cos }^{2}x-1$

We have,

${\cos }^{4}x-{\sin }^{4}x$

$\Rightarrow ({\cos }^{2}x{)}^2-({\sin }^{2}x{)}^2$

$\Rightarrow ({\cos }^{2}x-{\sin }^{2}x)({\cos }^{2}x+{\sin }^{2}x)$ $\because a^2-b^2=(a-b)(a+b)$

$\Rightarrow ({\cos }^{2}x-{\sin }^{2}x)\left(1\right)$ $\because {\cos }^{2}x+{\sin }^{2}x=1$

$\Rightarrow ({\cos }^{2}x-{\sin }^{2}x)$

$\Rightarrow 2{\cos }^{2}x-1$ $\because {\cos }^{2}x-{\sin }^{2}x=2co{s}^{2}x-1$

Hence, this is the answer.