prove these 2 trig identities and show the steps please.

Question

Tan^2a-sin^2a=tan^2asin^2a

Sinx(cscx-(cosx/cotx))=cos^2x?

Answer 1

(1).

The trigonometric equation is tan²(a) - sin²(a) = tan²(a) sin²(a).

Consider the left hand side expression is tan²(a) - sin²(a).

= [sin²(a)/cos²(a)] - sin²(a)              [ Since tan(θ) = sin(θ)/cos(θ) ]

= [sin²(a) - cos²(a) sin²(a)]/cos²(a)

= {sin²(a)[1 - cos²(a)]}/cos²(a)

= [sin²(a) sin²(a)]/cos²(a)                 [ Since sin²(θ) + cos²(θ) = 1 ]

= [sin²(a)/cos²(a)] sin²(a)

= tan²(a) sin²(a)

= right hand side expression.

Answer 2

(2).

The trigonometric equation is sin(x)[csc(x) - {cos(x)/cot(x)}] = cos²(x).

Consider the left hand side expression is sin(x)[csc(x) - {cos(x)/cot(x)}].

= sin(x)[1/sin(x) - {cos(x)*sin(x)/cos(x)}]  [ Since csc(θ) = 1/sin(θ) and cot(θ) = cos(θ)/sin(θ) ]

= sin(x)[1/sin(x) - sin(x)]

= sin(x)[1 - sin²(x)]/sin(x)

= [1 - sin²(x)]

= cos²(x)                 [ Since sin²(θ) + cos²(θ) = 1 ]

= right hand side expression.