Trigonometeric Equation

hint for 1st question -
use 2cosAcosB = cos(A+B)+cos(A-B) in rhs
use identity cos 4x = 2cos²(2x)-1 and cos 2x =cos²x-sin²x in rhs
simplify rhs and lhs.
convert terms to sin²2x and cos²2x then simple stuff.
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hint for q.3

check max. value of lhs
check min value of rhs
compare the two
finish it off :)

2)

2\left(tanx+\frac{1}{\sqrt{3}} \right)^2+\left(sinx-\frac{1}{2} \right)^2=0

The general soln. of which is

(2n+1)\pi-\frac{\pi}{6}