Question

If tan theta + sin theta = m & tan theta - sin theta , show that ( n square - n square ) = 4√mn

Answer 1

tanθ+sinθ=mtanθ-sinθ=n∴, m+n=tanθ+sinθ+tanθ-sinθ=2tanθm-n=tanθ+sinθ-tanθ+sinθ=2sinθmn=(tanθ+sinθ)(tanθ-sinθ)     =tan²θ-sin²θ∴, m²-n²=(m+n)(m-n)=2tanθ.2sinθ=4sinθtanθ4√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)
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