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Prove the Following Trigonometric Identities Cosec6θ = Cot6θ + 3 Cot2θ Cosec2θ + 1 - Mathematics

Prove the following trigonometric identities

cosec6θ = cot6θ + 3 cot2θ cosec2θ + 1

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Solution

We need to prove cosec6θ = cot6θ + 3 cot2θ cosec2θ + 1

Solving the L.H.S, we get

`cosec^6 theta = (cosec^2 theta)^3`

`= (1 + cot^2 theta)^3`     .......`(1 + cot^2 theta = cosec^2 theta)`

Further using the identity `(a + b)^3 = a^3 + b^3 + 3a^2b + 3ab^2`  we get

`(1 + cot^2 theta)^3 = 1 + cot^6 theta + 3(1)^2 (cot^2 theta) + 3(1) (cot^2 theta)^2`

`= 1 + cot^6 theta + 3 cot^2 theta + 3 cot^4 theta`

`= 1 + cot^6 theta + 3 cot^2 theta (1 + cot^2 theta)`

`= 1 + cot^6 theta + 3 cot^2 theta cosec^2 theta`    `(using 1 + cot^2 theta = cosec^2 theta)`

Hence proved.

 

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APPEARS IN

RD Sharma Class 10 Maths
Chapter 11 Trigonometric Identities
Exercise 11.1 | Q 32 | Page 44
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