"The number of solutions of sin7x + cos7x = 1, x$\\in$ [0, 4$\\pi$] is equal to"

"sin 7 x $\\le$ sin 2 x $\\le$ 1 ...... (1) and cos 7 x $\\le$ cos 2 x $\\le$ 1 ..... (2) also sin 2 x + cos 2 x = 1 $\\Rightarrow$ equality must hold for (1) & (2) $\\Rightarrow$ sin 7 x = sin 2 x & cos 7 x = cos 2 x $\\Rightarrow$ sin x = 0 & cos x = 1 or cos x = 0 & sin x = 1 $\\Rightarrow$ x = 0, 2$\\pi$, 4$\\pi$, ${\\pi \\over 2},{{5\\pi } \\over 2}$ $\\Rightarrow$ 5 solutions"

Medium MCQ +4 / -1 PYQ · JEE Mains 2021

The number of solutions of sin⁷x + cos⁷x = 1, x$\in$ [0, 4$\pi$] is equal to

A 11
B 7
C 5 Correct answer
D 9

Solution

sin7x $\le$ sin2x $\le$ 1 ...... (1) and cos7x $\le$ cos2x $\le$ 1 ..... (2) also sin2x + cos2x = 1 $\Rightarrow$ equality must hold for (1) & (2) $\Rightarrow$ sin7x = sin2x & cos7x = cos2x $\Rightarrow$ sin x = 0 & cos x = 1 or cos x = 0 & sin x = 1 $\Rightarrow$ x = 0, 2$\pi$, 4$\pi$, ${\pi \over 2},{{5\pi } \over 2}$ $\Rightarrow$ 5 solutions

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Subject: Mathematics · Chapter: Trigonometric Functions · Topic: Trigonometric Ratios and Identities

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The number of solutions of sin7x + cos7x = 1, x$\in$ [0, 4$\pi$] is equal to

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The number of solutions of sin⁷x + cos⁷x = 1, x$\in$ [0, 4$\pi$] is equal to