The simplification or the formula of 1+cot^{2}x is given as follows:

1+cot^{2}x = cosec^{2}x.

In this post, we will learn how to establish the formula of 1+cot^{2}x.

## Proof of 1+cot^{2}x Formula

**Question:** Prove the formula 1+cot^{2}x = cosec^{2}x.

**Solution:**

L.H.S. = 1+cot^{2}x

= $1+(\dfrac{\cos x}{\sin x})^2$ as cotx=cosx/sinx.

= $1+\dfrac{\cos^2 x}{\sin^2 x}$

= $\dfrac{\sin^2 x + \cos^2 x}{\sin^2 x}$

= $\dfrac{1}{\sin^2 x}$ as we know that sin^{2}x+cos^{2}x=1.

= $(\dfrac{1}{\sin x})^2$

= cosec^{2}x [∵ cosecx = 1/sinx]

= R.H.S.

So we have proven the formula of 1+cot^{2}x in terms of cosec which is given by 1+cot^{2}x = cosec^{2}x.

## Question-Answer

**Question:** Find the value of 1+cot^{2}45.

*Answer:*

In the above formula of 1+cot^{2}x = cosec^{2}x, we put x=45 degrees. So we get that

1+cot^{2}45 = cosec^{2}45 = (**√**2)^{2} = 2.

So the value of 1+cot^{2}45 is equal to 2.

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## FAQs

### Q1: What is the formula of 1+cot^{2}x?

Answer: The formula of 1+cot^{2}x is equal to 1+cot^{2}x=cosec^{2}x.

### Q2: What is the formula of 1+cot^{2}θ?

Answer: The formula of 1+cot^{2}θ is equal to 1+cot^{2}θ=cosec^{2}θ.