Question

express cot theta in terms of cosec theta​

Answer 1

Step-by-step explanation:

We know cosec ^2 theta = 1+ cot^2 theta

So cosec theta = √1+cot^2 theta

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Answer 2

cotθ = √(cosec²θ - 1) ( cotθ expressed in term of cosecθ) $\cot \theta = \sqrt{\csc^2{\theta}} -1 }$

Identities used:

• cotθ  = cosθ/sinθ
• cosecθ = 1/sinθ
• sin²θ + cos²θ = 1

Step 1:

Rewrite cotθ

cotθ  = cosθ/sinθ

Step 2:

Rewrite cosθ as √cos²θ

$\cot \theta = \frac{ \sqrt{\cos^2{\theta}} }{\sin \theta}$

Step 3:

Rewrite  cos²θ = 1 - sin²θ

$\cot \theta = \frac{ \sqrt{1 -\sin^2{\theta}} }{\sin \theta}$

Step 4:

Simplify the expression by taking sinθ under square root

$\cot \theta = \sqrt{ \frac{1}{\sin^2{\theta}} -1 }$

Step 4:

use cosecθ = 1/sinθ

$\cot \theta = \sqrt{\csc^2{\theta}} -1 }$

( cotθ expressed in term of cosecθ)   cotθ = √(cosec²θ - 1) $\cot \theta = \sqrt{\csc^2{\theta}} -1 }$

Note : cscθ and cosecθ  are the same thing