Verify the idenity 1 1 1sin2y 1 tan2y 2 1 tan2x 1 tan2x

Verify the idenity:

1) (1/ 1-sin²y) = 1 + tan²y

2) (1+ tan²x) / (1 - tan²x) = (1/ cos²x - sin²x)

3) (sin x + cos x) / (sec x + csc x ) = sinxcosx

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

5) (sin x-1) / (sin x+1) / (-cos^2x) / (sin x+1)^2

Solution

1) (1/ 1-sin²y)

=(1/ cos²y)

=(sin²y +cos²y)/cos²y.........................[since sin²y +cos²y =1]

=(sin²y/cos²y)  +(cos²y/cos²y)

=(siny/cosy)²  +1

=tan²y +1

=1+tan²y

2)(1+ tan²x) / (1 - tan²x)

=(1+ (sinx/cosx)²) / (1 - (sinx/cosx)²)

=[((cosx)²+ (sinx)²)/(cosx)²] / [((cosx)² - (sinx)²)/(cosx)²]

=[((cosx)²+ (sinx)²)] / [((cosx)² - (sinx)²)]

=1/((cosx)² - (sinx)²)

3)3) (sin x + cos x) / (sec x + csc x )

= (sin x + cos x) / ((1/cosx) + (1/sinx) )

= (sin x + cos x) / ((sinx+cosx)/(cosx sinx) )

=1/(1/(sinx cosx))

=cosxsinx