Question

If tan theta + sin theta= m and tan theta - sin theta =n then show that m²-n²=4√mn​

Answer 1

Step-by-step explanation:

Given, m=tanθ+sinθ,n=tanθ−sinθ

We need to show m

2

−n

2

=4

mn

Taking L.H.S.,

m

2

−n

2

=(tanθ+sinθ)

2

−(tanθ−sinθ)

2

=tan

2

θ+sin

2

θ+2tanθ.sinθ−tan

2

θ−sin

2

θ+2tanθ.sinθ

=4tanθ.sinθ

Now, taking R.H.S.,

4

mn

=4

(tanθ+sinθ)(tanθ−sinθ)

=4

tan

2

θ−sin

2

θ

=4

cos

2

θ

sin

2

θ

−sin

2

θ

=4

cos

2

θ

sin

2

θ(1−cos

2

θ)

=4

sin

2

θ.tan

2

θ

=4tanθ.sinθ

Therefore, L.H.S. = R.H.S.

Answer 2

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