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Mathematics

Prove the following identities :

sin A1 - cos A−cot A = cosec A\dfrac{\text{sin A}}{\text{1 - cos A}} - \text{cot A = cosec A}

Trigonometric Identities

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Answer

Solving L.H.S. of the equation :

⇒sin A1 - cos A−cot A⇒sin A1 - cos A−cos Asin A⇒sin2A−cos A(1 - cos A)sin A(1 - cos A)⇒sin2A−cos A + cos2Asin A(1 - cos A)\Rightarrow \dfrac{\text{sin A}}{\text{1 - cos A}} - \text{cot A} \\[1em] \Rightarrow \dfrac{\text{sin A}}{\text{1 - cos A}} - \dfrac{\text{cos A}}{\text{sin A}} \\[1em] \Rightarrow \dfrac{\text{sin}^2 A - \text{cos A(1 - cos A)}}{\text{sin A(1 - cos A)}} \\[1em] \Rightarrow \dfrac{\text{sin}^2 A - \text{cos A + cos}^2 A}{\text{sin A(1 - cos A)}}

By formula,

sin2 A + cos2 A = 1

⇒1 - cos Asin A(1 - cos A)⇒1sin A⇒cosec A.\Rightarrow \dfrac{\text{1 - cos A}}{\text{sin A(1 - cos A)}} \\[1em] \Rightarrow \dfrac{1}{\text{sin A}} \\[1em] \Rightarrow \text{cosec A}.

Since, L.H.S. = R.H.S.

Hence, proved that sin A1 - cos A−cot A = cosec A\dfrac{\text{sin A}}{\text{1 - cos A}} - \text{cot A = cosec A}.

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