Mathematics

Prove the following identities :
$\dfrac{\text{(cosec A - cot A)}^2 + 1}{\text{sec A (cosec A - cot A)}} = \text{2 cot A}$

Trigonometric Identities

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Answer

By formula,

cosec² A - cot² A = 1

Solving L.H.S. of the equation :

$\Rightarrow \dfrac{\text{(cosec A - cot A)}^2 + 1}{\text{sec A (cosec A - cot A)}} \\[1em] \Rightarrow \dfrac{\text{(cosec A - cot A)}^2 + \text{cosec}^2 A - \text{cot}^2 A}{\text{sec A (cosec A - cot A)}} \\[1em] \Rightarrow \dfrac{\text{(cosec A - cot A)}^2 + \text{(cosec A - cot A)(cosec A + cot A)}}{\text{sec A (cosec A - cot A)}} \\[1em] \Rightarrow \dfrac{\text{(cosec A - cot A)(cosec A - cot A + cosec A + cot A)}}{\text{sec A (cosec A - cot A)}} \\[1em] \Rightarrow \dfrac{\text{2 cosec A}}{\text{sec A}} \\[1em] \Rightarrow \dfrac{\dfrac{2}{\text{sin A}}}{\dfrac{1}{\text{cos A}}} \\[1em] \Rightarrow \dfrac{\text{2 cos A}}{\text{sin A}} \\[1em] \Rightarrow \text{2 cot A}.$

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

Hence, proved that $\dfrac{\text{(cosec A - cot A)}^2 + 1}{\text{sec A (cosec A - cot A)}} = \text{2 cot A}$ .

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Mathematics

Prove the following identities :
$\dfrac{\text{(cosec A - cot A)}^2 + 1}{\text{sec A (cosec A - cot A)}} = \text{2 cot A}$

Trigonometric Identities

Answer

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