Question

The total number of solution of the equation ${\sin }^{4}x+{\cos }^{4}x=\sin x\cos x$ in $[0,2\pi ]$ is :

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

The correct option is A $2$

$\begin{array}{c}{\sin }^{4}x+{\cos }^{4}x=\sin x.\cos x\\ \Rightarrow \left({\sin }^{2}x+{\cos }^{2}x\right)-2{\sin }^2.{\cos }^{2}x=\sin x.\cos x\\ \Rightarrow 1-\frac{{\sin }^{2}2x}{2}=\frac{\sin 2x}{2}\\ \Rightarrow {\sin }^{2}2x+\sin 2x-2=0\\ \Rightarrow \left(\sin 2x-1\right)\left(\sin 2x+2\right)=0\\ =\sin 2x=-2\left(notpossible\right)\\ =\sin 2x=1\\ =2x=\frac{\pi }{2},\frac{5\pi }{2}\\ =x=\frac{\pi }{4},\frac{5\pi }{4}\end{array}$