Verify that a sin x cos x2 sin2x 1 b 11costheta csc2thet

Verify that:

a) (sin x + cos x)^2 = sin(2x) + 1

b) 1/(1+cos(theta)) = csc^2(theta) - csc(theta)cot(theta)

Solution

a) (sin x + cos x)^2 = sin(2x) + 1

LHS = sin^2 x + cos^2 x + 2*sin x*cos x

        = 1 + sin2x = RHS

b)    1/(1+cos ) = csc^2() - csc()cot()

RHS = (1/sin^2 ) – (1/sin )(cos /sin )

        = (1/sin^2 ) - (cos /sin^2 )

        = (1 - cos )/sin^2

         = (1 – cos^2 )/[ sin^2 *(1 + cos )]

         = 1/[(1 + cos )] = LHS