Question

Find the value of
cos⁻¹(1/2) + 2sin⁻¹(1/2)

Answer 1

As we know that

$\cos( \frac{\pi}{3} ) = \frac{1}{2} \\ \\ \sin(\frac{\pi}{6} ) = \frac{1}{2}$
and the principal value branch of cos⁻¹(x) and sin⁻¹(x) are [0,π] and [ -π/2, π/2] respectively

${cos}^{ - 1} cos \: ( \frac{\pi}{3} ) + 2 {sin}^{ - 1} sin( \frac{\pi}{6} ) \\ \\ = \frac{\pi}{3} + 2 \times \frac{\pi}{6} \\ \\ = \frac{\pi}{3} + \frac{\pi}{3} \\ \\ = \frac{2\pi}{3}$

as we know inverse trigonometric functions cancels the trigonometric functions if the angle lie between it's principal value branch.