The simplification of the expression 1-cos^{2}x is equal to sin^{2}x. In this post, we will learn how to find the formula of 1-cos^2x.

## 1-cos^{2}x Formula

The formula of 1-cos^{2}x is given below:

1-cos^{2}x = sin^{2}x

## 1-cos^{2}x = sin^{2}x Proof

To prove $1-\cos^2x =\sin^2 x$, we will follow the below steps:

**Step 1:** At first, we will apply the Pythagorean trigonometric identity which is provided below.

$\sin^2 x +\cos^2 x=1$ **…(I)**

**Step 2:** Next, we will substitute the above value of 1 in the expression $1-\cos^2 x$. This will give us

$1-\cos^2 x= (\sin^2 x +\cos^2 x)-\cos^2 x$

$=\sin^2 x +\cos^2 x-\cos^2x$

$=\sin^2 x$

So the simplification of $1-\cos^2 x$ is equal to $\sin^2 x$. In other words,

1-cos^{2}x = sin^{2}x |

**Note** that if we substitute θ in place of x, we will get the formula of 1-cos^{2}θ which is provided below:

$1-\cos^2 \theta=\sin^2 \theta$.

**Also Read:**

**Values of sin 15, cos 15, tan 15**

**Values of sin 75, cos 75, tan 75**

## Question-Answer on 1-sin2x Formula

**Question 1:** Find the value of $1-\cos^2 45^\circ$

*Answer:*

From the above, we have $1-\cos^2 x=\sin^2 x$.

Let us put $x=45^\circ$.

Thus, we get that

$1-\cos^2 45^\circ$

$=\sin^2 45^\circ$

$=(\dfrac{1}{\sqrt{2}})^2$

$=1/2$ as we know that $\sin 45^\circ =\dfrac{1}{\sqrt{2}}$.

So the value of $1-\cos^2 45^\circ$ is equal to $1/2$.

## FAQs

**Q1: What is the formula of 1-cos ^{2}x?**

**Answer:** The formula of 1-cos^{2}x is given by 1-cos^{2}x=sin^{2}x.

**Q2: What is the formula of 1-cos ^{2}θ?**

**Answer:** The formula of 1-cos^{2}θ is given as follows: 1-cos^{2}θ=sin^{2}θ.