Question

if tan theta + sin theta m and tan theta - sin theta = n , show that m2 - n2 = 4 root mn

Answer 1

I hope it will help you.

Answer 2

tanθ+sinθ=m

tanθ-sinθ=n

∴, m+n=tanθ+sinθ+tanθ-sinθ=2tanθ

m-n=tanθ+sinθ-tanθ+sinθ=2sinθ

mn=(tanθ+sinθ)(tanθ-sinθ)

    =tan²θ-sin²θ

∴, m²-n²

=(m+n)(m-n)

=2tanθ.2sinθ

=4sinθtanθ

4√mn

=4√(tan²θ-sin²θ)

=4√(sin²θ/cos²θ-sin²θ)

=4√sin²θ(1/cos²θ-1)

=4sinθ√(1-cos²θ)/cos²θ

=4sinθ/cosθ√sin²θ [∵, sin²θ+cos²θ=1]

=4sinθtanθ

∴, LHS=RHS (proved