The number of distinct solutions of the equation `5/4cos^(2)2x + cos^4 x + sin^4 x+cos^6x+sin^6 x =2` in the interval `[0,2pi] ` is

Options

A. 4
B. 6
C. 8
D. 5

Answer

C

Explanation

We have,
`(5)/(4)"cos"^(2)2x + "cos"^(4)x + "sin"^(4)x + "cos"^(6)x + "sin"^(6)x = 2`
`rArr (5)/(4) "cos"^(2) 2x + ("cos"^(2) x + "sin"^(2)x)^(2) - 2"cos"^(2)x "sin"^(2)x + ("cos"^(2)x + "sin"^(2)x)^(3)-3"cos"^(2)x"sin"^(2)x("cos"^(2)x + "sin"^(2)x) =2`
`(5)/(4)"cos"^(2)2x + 1-(1)/(2)"sin"^(2)2x +1-(3)/(4)"sin"^(2)2x =2`
`rArr (5)/(4)"cos"^(2)2x- (5)/(4)"sin"^(2)2x =0`
`rArr "tan"^(2)2x = 1`
`"tan" 2x = +-1, x in [0, 2pi]`
`rArr 2x = (pi)/(4), (3pi)/(4), (5pi)/(4), (7pi)/(4), (9pi)/(4), (11pi)/(4), (13pi)/(4), (15pi)/(4)`
`rArr x = (pi)/(8), (3pi)/(8), (5pi)/(8), (7pi)/(8), (9pi)/(8), (11pi)/(8), (13pi)/(8), (15pi)/(8)`
Clearly, these are 8 solutions.