cos^2(θ+pi/6)= 1/2

Solution

cos^{2} (θ + \frac{π}{6}) = \frac{1}{2}

θ = 2 πn + \frac{π}{12}, θ = 2 π + 2 πn - \frac{5 π}{12}, θ = 2 πn + \frac{7 π}{12}, θ = 2 πn - \frac{11 π}{12}

Solution steps

cos^{2} (θ + \frac{π}{6}) = \frac{1}{2}

Solve by substitution

cos (θ + \frac{π}{6}) = \frac{1}{2}, cos (θ + \frac{π}{6}) = - \frac{1}{2}

cos (θ + \frac{π}{6}) = \frac{1}{2} : θ = 2 πn + \frac{π}{12}, θ = 2 π + 2 πn - \frac{5 π}{12}

cos (θ + \frac{π}{6}) = - \frac{1}{2} : θ = 2 πn + \frac{7 π}{12}, θ = 2 πn - \frac{11 π}{12}

Combine all the solutions

θ = 2 πn + \frac{π}{12}, θ = 2 π + 2 πn - \frac{5 π}{12}, θ = 2 πn + \frac{7 π}{12}, θ = 2 πn - \frac{11 π}{12}

2 sin (2 x) = 2 sin (x)

tan (x) = \frac{1}{7}

cos^{2} (x) = 0.1

cos (x) = 0.184

y, x = cos^{2} (y)