Question

The number of solutions of the equation 1 + sin⁴x = cos³3x, x ε [-5π/2, 5π/2] is :
(A) 3 (B) 4
(C) 5 (D) 7

Answer 1

your answer is

step by step

Answer 2

The number of solutions of the equation 1 + sin⁴x = cos³3x, x ε [-5π/2, 5π/2] is:

The correct equation is

1 + sin⁴x = cos²3x

Given,

$1 + sin^4x = cos^23x\\\\1 + sin^4x = 1 - sin^23x\\\\sin^4x = - sin^23x\\\\sin^23x + sin^4x = 0\\\\(3sin x - 4sin ^3x)^2 + sin^4x = 0\\\\9sin^2x + 16sin^6x - 24sin^4x+ sin^4x = 0\\\\9sin^2x + 16sin^6x - 23sin^4x = 0\\\\sin^2x ( 9 + 16sin^4x - 23sin^2x) = 0\\\\sin^2x = 0 \Rightarrow x = n\pi\\or\\9 + 16sin^4x - 23sin^2x = 0 \Rightarrow \dfrac{\sqrt{47}}{8} \pm \dfrac{1}{8}i, -\dfrac{\sqrt{47}}{8} \pm \dfrac{1}{8}i$

Therefore a total of 5 solutions

Option (C) is correct.