The number of solution of sin7 x + cos7 x = 1, x [0, 4π] is equal to :

Engineering

Mathematics

Trigonometric Inequalities

Question

The number of solution of sin⁷ x + cos⁷ x = 1, x $\in$ [0, 4π] is equal to :

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sin²x + cos²x = 1, sin²x $\leq$ 1 and cos²x $\leq$ 1

sin⁷x $\leq$ sin²x

cos⁷x $\leq$ cos²x

so, sin⁷x + cos⁷x $\leq$ 1

sin⁷x + cos⁷x = 1 when sin⁷x = sin²x & cos⁷x = cos²x

Case-1 : sin x = 0, cos x = 1 ⇒ x = 0 , 2π, 4π

Case-2 : sin x = 1, cos x = 0 ⇒ $x = \frac{π}{2}, \frac{5 π}{2}$

Total number of solutions = 5

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