Question

Prove sinθ+cosθ/sinθ−cosθ+sinθ−cosθ/sinθ+cosθ=2/sin2θ−cos2θ=2/2sin2θ−1=2sec2θ/tan2 θ−1

Answer 1

Answer:

ANSWER

sinθ−cosθ

sinθ+cosθ

+

sinθ+cosθ

sinθ−cosθ

=

tan

2

θ−1

2sec

2

θ

L.H.S

=

(sinθ+cosθ)(sinθ−cosθ)

(sinθ+cosθ)

2

+(sinθ−cosθ)

2

=

sin

2

θ−cos

2

θ

sin

2

θ+cos

2

θ+2sinθcosθ+sin

2

θ+cos

2

θ−2sinθcosθ

=

sin

2

θ−cos

2

θ

sin

2

θ+cos

2

θ+sin

2

θ+cos

2

θ

=

sin

2

θ−cos

2

θ

2sin

2

θ+2cos

2

θ

=

(sin

2

θ−cos

2

θ)

2(sin

2

θ+cos

2

θ)

Now, RHS

=

tan

2

θ−1

2sec

2

θ

=

cos

2

θ

sin

2

θ

−1

2

cos

2

θ

1

=

sin

2

θ−cos

2

θ/cos

2

θ

2/cos

2

θ

=

sin

2

θ−cos

2

θ

2

∴ L.H.S=R.H.S