Verify the following trigonometric identities Show any work

Verify the following trigonometric identities. Show any work necessary.

2. a) sinx - sinxcos^2x = sin^3x

LHS : sinx - sinxcos^2x = sinx( 1 -cos^2x)

= sinx( sin^2x)

= sin^3x

= RHS

b) ( sin^2x +4sinx +3)/cos^2x = ( 3+sinx)/( 1-sinx)

LHS : ( sin^2x +4sinx +3)/cos^2x

factorise numerator: sin^2x + 4sinx +3 = sin^2x +3sinx +sinx +3

= sinx( sinx +3) +1(sinx +3)

= (1+sinx)( sinx +3)

( sin^2x +4sinx +3)/cos^2x = (1+sinx)( sinx +3)/( 1-sin^2x)

= (1+sinx)( sinx +3)//( 1+sinx)(1 - sinx)

cancelling common factors in numerator and denominator:

= (sinx +3)/( 1- sinx)

= RHS

c) tan^2x/( 1+tan^2x ) = sin^2x

LHS: tan^2x/( 1+tan^2x )

using trigonometric identities: 1+tan^2x = sec^2x

tan^2x/sec^2x = sin^2x

= RHS