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Mathematics

Prove the following identities :

1 + sin A1 - sin A=cosec A + 1cosec A - 1\dfrac{\text{1 + sin A}}{\text{1 - sin A}} = \dfrac{\text{cosec A + 1}}{\text{cosec A - 1}}

Trigonometric Identities

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Answer

Solving R.H.S. of the equation :

⇒cosec A + 1cosec A - 1⇒1sin A+11sin A−1⇒1 + sin Asin A1 - sin Asin A⇒(1 + sin A) × sin A(1 - sin A) × sin A⇒1 + sin A1 - sin A.\Rightarrow \dfrac{\text{cosec A + 1}}{\text{cosec A - 1}} \\[1em] \Rightarrow \dfrac{\dfrac{1}{\text{sin A}} + 1}{\dfrac{1}{\text{sin A}} - 1} \\[1em] \Rightarrow \dfrac{\dfrac{\text{1 + sin A}}{\text{sin A}}}{\dfrac{\text{1 - sin A}}{\text{sin A}}} \\[1em] \Rightarrow \dfrac{\text{(1 + sin A) × sin A}}{\text{(1 - sin A) × sin A}} \\[1em] \Rightarrow \dfrac{\text{1 + sin A}}{\text{1 - sin A}}.

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

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

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