cos^2(x)=sin^2(x/2)

Solution

cos^{2} (x) = sin^{2} (\frac{x}{2})

x = - \frac{π}{3} - \frac{8 πn}{3}, x = - \frac{5 π}{3} - \frac{8 πn}{3}, x = \frac{8 πn}{3} + \frac{π}{3}, x = \frac{5 π}{3} + \frac{8 πn}{3}

Solution steps

cos^{2} (x) = sin^{2} (\frac{x}{2})

Subtract

sin^{2} (\frac{x}{2})

from both sides

cos^{2} (x) - sin^{2} (\frac{x}{2}) = 0

Rewrite using trig identities

- cos^{2} (\frac{- x + π}{2}) + cos^{2} (x) = 0

Factor

- cos^{2} (\frac{- x + π}{2}) + cos^{2} (x) : (cos (x) + cos (\frac{- x + π}{2})) (cos (x) - cos (\frac{- x + π}{2}))

(cos (x) + cos (\frac{- x + π}{2})) (cos (x) - cos (\frac{- x + π}{2})) = 0

Solving each part separately

cos (x) + cos (\frac{- x + π}{2}) = 0 or cos (x) - cos (\frac{- x + π}{2}) = 0

cos (x) + cos (\frac{- x + π}{2}) = 0 : x = - \frac{π}{3} - \frac{8 πn}{3}, x = - \frac{5 π}{3} - \frac{8 πn}{3}

cos (x) - cos (\frac{- x + π}{2}) = 0 : x = \frac{8 πn}{3} + \frac{π}{3}, x = \frac{5 π}{3} + \frac{8 πn}{3}

Combine all the solutions

x = - \frac{π}{3} - \frac{8 πn}{3}, x = - \frac{5 π}{3} - \frac{8 πn}{3}, x = \frac{8 πn}{3} + \frac{π}{3}, x = \frac{5 π}{3} + \frac{8 πn}{3}

sin (x + \frac{π}{4}) = - \frac{2}{2}

1 = tan (θ)

2 cos (x) - 4 = 0

sec (x) = 1 - tan (x)

2 sin (x) sin (3 x) = 1, 0 \leq x \leq 2 π