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Prove that : tan2 A + cot2 A + 2 = sec2 A. cosec2 A

Trigonometric Identities

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Answer

To prove:

tan2 A + cot2 A + 2 = sec2 A. cosec2 A

Solving L.H.S. of the above equation :

⇒tan2A+cot2A+2⇒sin2Acos2A+cos2Asin2A+2⇒sin4A+cos4A+2 sin2A cos2Asin2A cos2A⇒(sin2A+cos2A)2sin2A cos2A⇒(1)2sin2A cos2A⇒1sin2A×1cos2A⇒cosec2A. sec2A.\phantom{\Rightarrow} \text{tan}^2 A + \text{cot}^2 A + 2 \\[1em] \Rightarrow \dfrac{\text{sin}^2 A}{\text{cos}^2 A} + \dfrac{\text{cos}^2 A}{\text{sin}^2 A} + 2 \\[1em] \Rightarrow \dfrac{\text{sin}^4 A + \text{cos}^4 A + \text{2 sin}^2 A \text{ cos}^2 A}{\text{sin}^2 A\text{ cos}^2 A} \\[1em] \Rightarrow \dfrac{(\text{sin}^2 A + \text{cos}^2 A)^2}{\text{sin}^2 A \text{ cos}^2 A} \\[1em] \Rightarrow \dfrac{(1)^2}{\text{sin}^2 A \text{ cos}^2 A} \\[1em] \Rightarrow \dfrac{1}{\text{sin}^2 A} \times \dfrac{1}{\text{cos}^2 A} \\[1em] \Rightarrow \text{cosec}^2 A. \text{ sec}^2 A.

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

Hence, proved that tan2 A + cot2 A + 2 = sec2 A. cosec2 A

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