Write the trigonometric expression as an algebraic expressio

Write the trigonometric expression as an algebraic expression in the variable u. tan(sin^-1 u) Establish the identity. 1 - cos theta/1 + cos theta = (csc theta - cot theta)^2

Solution

1.

tan (sin^-1 u) can be written as tan A,
where A = sin^-1 u => sin A = u = u/1

we know that sin of an angle is perpendicular/hypotenuse,

so by pythagorean theorem,
base = (1² – u²) = (1 – u²)

Thus,          tan (sin^-1 u) = u / (1 – u²)

2.

Here we have on LHS = (1 – cos) / (1 + cos)

Multiplying numerator and denominator by (1 – cos), we get

LHS = (1 – cos)² / (1 + cos)*(1 – cos)

            = (1 + cos² – 2cos) / (1 - cos²)

            = (1 + cos² – 2cos) / (sin²)

            = cosec² + cot² – 2cot .cosec

            = (cosec - cot )²

            = RHS