The number of solutions of sin ^{7} x+cos ^{7 | vedclass

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Question

$\sin ^{7} x \leq \sin ^{2} x \leq 1....(1)$

$	ext { and } \cos ^{7} x \leq \cos ^{2} x \leq 1....(2)$

$	ext { also } \sin ^{2} x+\cos ^{2} x=

Vedclass · Accepted Answer

$5$

Vedclass · Answer

$\sin ^{7} x \leq \sin ^{2} x \leq 1....(1)$

$	ext { and } \cos ^{7} x \leq \cos ^{2} x \leq 1....(2)$

$	ext { also } \sin ^{2} x+\cos ^{2} x=1$

$\Rightarrow 	ext { equality must hold for }(1) \,\,(2)$

$\Rightarrow \sin ^{7} x=\sin ^{2} x \,\, \cos ^{7}=\cos ^{2} x$

$\Rightarrow \sin x=0\, \, \cos x=1 	ext { or }$

$\cos x=0\,  \,\sin x=1$

$\Rightarrow x=0,2 \pi, 4 \pi, \frac{\pi}{2}, \frac{5 \pi}{2}$

$\Rightarrow 5 	ext { solutions }$

The number of solutions of $\sin ^{7} x+\cos ^{7}=1, x \in[0,4 \pi]$ is equal to :

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