Question

If tan theta + sin theta = m and tan theta – sin theta = n show that m^ 2 – n^ 2 = 4 root mn

Answer 1

HELLO DEAR,

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--------(1)

-----------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-----------(1)

from--(1) and----(2)

m²-n² = 4√mn

Hope helps.......

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