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Mathematics

Prove that :

(sec θ − cos θ)(cosec θ − sin θ) = sin θ cos θ

Trigonometric Identities

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Answer

Solving,

⇒(1cos⁡θ−cos⁡θ)(1sin⁡θ−sin⁡θ)⇒(1−cos⁡2θcos⁡θ)(1−sin⁡2θsin⁡θ)⇒(sin⁡2θcos⁡θ)(cos⁡2θsin⁡θ)⇒sin⁡θcos⁡θ.\Rightarrow \Big(\dfrac{1}{\cos θ} - \cos θ\Big) \Big(\dfrac{1}{\sin θ} - \sin θ\Big) \\[1em] \Rightarrow \Big(\dfrac{1 - \cos^2 θ}{\cos θ}\Big) \Big(\dfrac{1 - \sin^2 θ}{\sin θ}\Big) \\[1em] \Rightarrow \Big(\dfrac{\sin^2 θ}{\cos θ}\Big) \Big(\dfrac{\cos^2 θ}{\sin θ}\Big) \\[1em] \Rightarrow \sin θ \cos θ.

Hence, proved (sec θ − cos θ)(cosec θ − sin θ) = sin θ cos θ.

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