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# Prove the Following Trigonometric Identities Tan2 A + Cot2 A = Sec2 A Cosec2 A − 2 - CBSE Class 10 - Mathematics

#### Question

Prove the following trigonometric identities

tan2 A + cot2 A = sec2 A cosec2 A − 2

#### Solution

In the given question, we need to prove tan2 A + cot2 A = sec2 A cosec2 A − 2

Now using tan theta = sin theta/cos theta and cot theta = cos theta/sin theta in LHS we get

tan^2 A + cot^2  A = sin^2 A/cos^2 A + cos^2 A/sin^2 A

= (sin^4 A + cos^4 A)/(cos^2 A sin^2 A)

= ((sin^2 A)^2 + (cos^2 A)^2)/(cos^2 A sin^2 A)

Further, using the identity a^2 + b^2 = (a + b)^2 - 2ab we get

((sin^2 A)^2 + (cos^2 A)^2)/(cos^2 A sin^2 A) = ((sin^2 A + cos^ A)^2 - 2 sin^2 A cos^2 A)/(sin^2 A cos^2 A)

= ((1)^2 - 2sin^2 A cos^2 A)/(sin^2 A cos^2 A)

= 1/(sin^2 A cos^2 A) - (2 sin^2 A cos^2 A)/(sin^2 A cos^2 A

= cosec^2 A sec^2 A - 2

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

Hence proved.

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Solution Prove the Following Trigonometric Identities Tan2 A + Cot2 A = Sec2 A Cosec2 A − 2 Concept: Trigonometric Identities.
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