What is the exact value of sin(cos−1(−√3/2))+tan−1(sin(−π/2))?

Let's break down the expression: cos ⁡ − 1 ( − 3 / 2 ) cos −1 (− 3 ​ /2): This is the inverse cosine of − 3 / 2 − 3 ​ /2, which corresponds to an angle of −  / 3 −π/3 radians. sin ⁡ ( cos ⁡ − 1 ( − 3 / 2 ) ) sin(cos −1 (− 3 ​ /2)): This is the sine of the angle −  / 3 −π/3, which is − 3 / 2 − 3 ​ /2. tan ⁡ − 1 ( sin ⁡ ( −  / 2 ) ) tan −1 (sin(−π/2)): This is the inverse tangent of sin ⁡ ( −  / 2 ) Cat Needs Best on sin(−π/2), which is undefined. Now, let's substitute these values into the original expression: Expression = ( − 3 / 2 ) + undefined Expression=(− 3 ​ /2)+undefined Since the second term is undefined, the entire expression is undefined. Therefore, none of the given options (A, B, C, D) matches the value of the expression. It's possible there might be a mistake in the given expression or options.

sin (arcos (-sqrt (3) / 2)) =  sin (5/6 pi)  =  1/2

arctan ( sin (-pi/2))  =  arctan (-1)  = 7pi/4  =  -pi/4

1/2 - pi/4  =

[ 2  - pi ] /  4