2(arcsin(x))^2+(2-pi)arcsin(x)-pi=0

Solution

2 (arcsin (x))^{2} + (2 - π) arcsin (x) - π = 0

x = sin (\frac{- 2 + π + π ^{2} + 4 π + 4}{4}), x = sin (\frac{- 2 + π - π ^{2} + 4 π + 4}{4})

Solution steps

2 (arcsin (x))^{2} + (2 - π) arcsin (x) - π = 0

Solve by substitution

arcsin (x) = \frac{- 2 + π + π ^{2} + 4 π + 4}{4}, arcsin (x) = \frac{- 2 + π - π ^{2} + 4 π + 4}{4}

arcsin (x) = \frac{- 2 + π + π ^{2} + 4 π + 4}{4} : x = sin (\frac{- 2 + π + π ^{2} + 4 π + 4}{4})

arcsin (x) = \frac{- 2 + π - π ^{2} + 4 π + 4}{4} : x = sin (\frac{- 2 + π - π ^{2} + 4 π + 4}{4})

Combine all the solutions

x = sin (\frac{- 2 + π + π ^{2} + 4 π + 4}{4}), x = sin (\frac{- 2 + π - π ^{2} + 4 π + 4}{4})

cos^{2} (x) + 3 sin (- x) - 2 = 0

cos (x) = - 1, 0 < x < 2 π

\frac{1}{1 + cos ( x )} + \frac{1}{1 - cos ( x )} = 32

sin^{2} (x) = 14

sin (2 x) \cdot cos (3 x) = 0