Question

Prove each of these identities.

Answer 1

Answer:

Identity (a) can be re-written as

$sec x\ cosec x - cot x = tan x$

which we already proven in another question, while for idenity (b)

$(A)\frac 1 {sin x} -sin x = cos x \frac{cos x}{sin x}\\\\(B)\frac {1-sin^2x}{sin x} = \frac {cos^2x} {sin x} \\\\(C) \frac {cos^2x}{sin x} =\frac {cos^2x} {sin x}$

step A is simply expressing each function in terms of sine and cosine.

step B is adding the terms on the LHS while multiplying the one on RHS.

step C is replacing the term on the numerator with the equivalent from the pithagorean identity $cos^2x + sin^2x = 1$