Question

If \(\sin x-\cos x=0\), then what is the value of \(\sin ^4 x+\cos ^4 x ?\)

Answer 1

Given \(\sin x-\cos x=0\)
\[
\begin{aligned}
& \Rightarrow \sin x=\cos x \\
& \Rightarrow \tan x=1 \\
& \therefore \tan x=\tan \frac{\pi}{4} \quad \text { as } \tan 45^{\circ}=1 \\
& \Rightarrow x=\frac{\pi}{4}
\end{aligned}
\]
\[
\begin{aligned}
\therefore \sin ^4 x+\cos ^2 x & =\sin ^4 \frac{\pi}{4}+\cos ^4 \frac{\pi}{4} \\
& =\left(\frac{1}{\sqrt{2}}\right)^4+\left(\frac{1}{\sqrt{2}}\right)^4 \\
& =\frac{1}{4}+\frac{1}{4}
\end{aligned}
\]
 \[\sin ^4 x+\cos ^2 x=\frac{1}{2}\]