Tan theta -cot theta / sin theta cos theta =tan square theta -cot square theta
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tanθ-cotθ/sinθcosθ
=(sinθ/cosθ-cosθ/sinθ)/sinθcosθ
=[sin²θ-cos²θ)/sinθcosθ]/sinθcosθ
=(sin²θ-cos²θ)/sin²θcos²θ
=sin²θ/sin²θcos²θ-cos²θ/sin²θcos²θ
=1/cos²θ-1/sin²θ
=sec²θ-cosec²θ
=1+tan²θ-(1+cot²θ) [∵, sec²θ-tan²θ=1, cosec²θ-cot²θ=1]
=1+tan²θ-1-cot²θ
=tan²θ-cot²θ (Proved)
=(sinθ/cosθ-cosθ/sinθ)/sinθcosθ
=[sin²θ-cos²θ)/sinθcosθ]/sinθcosθ
=(sin²θ-cos²θ)/sin²θcos²θ
=sin²θ/sin²θcos²θ-cos²θ/sin²θcos²θ
=1/cos²θ-1/sin²θ
=sec²θ-cosec²θ
=1+tan²θ-(1+cot²θ) [∵, sec²θ-tan²θ=1, cosec²θ-cot²θ=1]
=1+tan²θ-1-cot²θ
=tan²θ-cot²θ (Proved)
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