Question

Prove that : cosec 6 theta = cot 6 theta + 3 cot2 theta × sec 2 theta +1

Answer 1

$\text{cosec}^6 \theta = \text{cot}^6 \theta + 3\text{cot}^2\theta \times \text{cosec}^2 \theta + 1$ is proved

Step-by-step explanation:

To Prove :

$\text{cosec}^6 \theta = \text{cot}^6 \theta + 3\text{cot}^2\theta \times \text{cosec}^2 \theta + 1$

We know that,

$\text{cosec}^2 \theta = \text{cot}^2 \theta + 1 \ \ -------> (1)$

Cubing on both sides in the equation we get,

$\text{(cosec}^2 \theta)^3 = \text{(cot}^2 \theta)^3 + 1 \ \ ------> (2)$

Also, we know that (a+b)³ = a³ + b³+ 3 ab (a+b)

$(\text{cot}^2 \theta + 1)^3 \ = \text{(cot}^2 \theta)^3 + 1^3 + 3\text{(cot}^2 \theta)(\text{cot}^2 \theta + 1) \ \ ------> (3)$

Substitute the above equation (3) in equation in (2)

$(\text{cosec} \theta )^6 \ = \text{(cot}^2 \theta)^3 + 1^3 + 3\text{(cot}^2 \theta)(\text{cot}^2 \theta + 1)$

From equation (1),

$(\text{cosec} \theta )^6 \ = \text{(cot}^6 \theta) + 3\text{(cot}^2 \theta)(\text{cosec}^2 \theta) + 1$

Hence the given condition is proved.

To Learn More:

1. If tan theta + 1/tan theta = 2, find the value of tan square theta + 1/tan square theta

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2. Cos 2 theta/(1-tan theta)+sin 3 theta/(sin theta - cos theta)=1+sin theta.cos theta

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