The number of values of `theta` in the interval `(-(pi)/(2), (pi)/(2))` such that `theta ne (n pi)/(5) " for n" = 0, +-1, +-2 " and tan" theta = "cot"5theta " as well as sin" 2 theta = "cos" 4theta`, is

Options

A. 4
B. 5
C. 7
D. 3

Answer

D

Explanation

`"tan" theta = "cot" 5theta`
`rArr "tan"theta-"cot"5theta = 0`
`rArr "cos"5theta"cos"theta - "sin"5theta "sin"theta = 0 rArr "cos"6theta =0`
`rArr 6 theta = +- (pi)/(2), +- (3pi)/(2), +- (5pi)/(2)`
`rArr theta = +- (pi)/(12), +- (pi)/(4), +- (5pi)/(12)`
Now, `"sin" 2theta = "cos" 4 theta`
`rArr "sin" 2 theta = 1 -2"sin"^(2) 2theta`
`rArr 2"sin"^(2) 2theta + "sin" 2 theta -1 =0`
`rArr (2"sin" 2theta -1)("sin" 2theta + 1) = 0`
`rArr "sin"2theta = (1)/(2) "or", 2theta = (pi)/(2)`
`rArr theta = (pi)/(12), (5pi)/(12)"or" theta = (pi)/(4)`
Clearly, the common values of `theta` are `- (pi)/(4), (pi)/(12) " and " (5pi)/(12)`
Hence, there are 3 values of `theta` satisfying the given equations.