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Mathematics

Prove the following identities :

cot2 A - cos2 A = cos2 A. cot2 A

Trigonometric Identities

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Answer

Solving L.H.S. of the equation :

⇒cot2A−cos2A⇒cos2Asin2A−cos2A⇒cos2A(1sin2A−1)⇒cos2A×(1−sin2Asin2A)⇒cos2Asin2A×1 - sin2A⇒cot2A×(1 - sin2A)\Rightarrow \text{cot}^2 A - \text{cos}^2 A \\[1em] \Rightarrow \dfrac{\text{cos}^2 A}{\text{sin}^2 A} - \text{cos}^2 A \\[1em] \Rightarrow \text{cos}^2 A\Big(\dfrac{1}{\text{sin}^2 A} - 1\Big) \\[1em] \Rightarrow \text{cos}^2 A \times \Big(\dfrac{1 - \text{sin}^2 A}{\text{sin}^2 A}\Big) \\[1em] \Rightarrow \dfrac{\text{cos}^2 A}{\text{sin}^2 A} \times \text{1 - sin}^2 A \\[1em] \Rightarrow \text{cot}^2 A \times (\text{1 - sin}^2 A)

By formula,

1 - sin2 A = cos2 A.

⇒cot2A.cos2A\Rightarrow \text{cot}^2 A .\text{cos}^2 A

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

Hence, proved that cot2 A - cos2 A = cos2 A. cot2 A

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