Question

Find the value of tan¯¹ [2 cos (2sin-1 1/2) ]​

Answer 1

• The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin(x), arccos(x), arctan(x), etc.

$\sf{\arctan \left(2\cos \left(2\arcsin \left(\dfrac{1}{2}\right)\right)\right)}$

$\sf{\displaystyle=\arctan \left(2\left(-1+2\cos ^2\left(\arcsin \left(\frac{1}{2}\right)\right)\right)\right)}$

$\sf{\displaystyle=\arctan \left(2\left(-1+2\left(\sqrt{1-\left(\frac{1}{2}\right)^2}\right)^2\right)\right)}$

$\sf{\displaystyle=\arctan \left(2\left(\frac{3}{2}-1\right)\right)}$$=\arctan \left(2\cdot \frac{1}{2}\right)$

$\large\boxed{\sf{ =arctan \left(1\right)}}$