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Mathematics

Prove that :

cot A - tan A = 2 cos2A−1sin A cos A\dfrac{\text{2 cos}^2 A - 1}{\text{sin A cos A}}

Trigonometric Identities

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Answer

Solving R.H.S. of the equation :

⇒2 cos2A−1sin A cos A⇒cos2A−1+cos2Asin A cos A⇒cos2A−(1 - cos2A)sin A cos A\Rightarrow \dfrac{\text{2 cos}^2 A - 1}{\text{sin A cos A}} \\[1em] \Rightarrow \dfrac{\text{cos}^2 A - 1 + \text{cos}^2 A}{\text{sin A cos A}} \\[1em] \Rightarrow \dfrac{\text{cos}^2 A - \text{(1 - cos}^2 A)}{\text{sin A cos A}}

By formula,

1 - cos2 A = sin2 A

⇒cos2A−sin2Asin A cos A⇒cos2Asin A cos A−sin2Asin A cos A⇒cos Asin A−sin Acos A⇒cot A - tan A.\Rightarrow \dfrac{\text{cos}^2 A - \text{sin}^2 A}{\text{sin A cos A}} \\[1em] \Rightarrow \dfrac{\text{cos}^2 A}{\text{sin A cos A}} - \dfrac{\text{sin}^2 A}{\text{sin A cos A}} \\[1em] \Rightarrow \dfrac{\text{cos A}}{\text{sin A}} - \dfrac{\text{sin A}}{\text{cos A}} \\[1em] \Rightarrow \text{cot A - tan A}.

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

Hence, proved that cot A - tan A = 2 cos2A−1sin A cos A\dfrac{\text{2 cos}^2 A - 1}{\text{sin A cos A}}.

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