Simplify 1-cos^2x | 1-cos^2x Formula

The simplification of the expression 1-cos²x is equal to sin²x. In this post, we will learn how to find the formula of 1-cos^2x.

1-cos²x Formula

The formula of 1-cos²x is given below:

1-cos²x = sin²x

1-cos²x = sin²x 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-cos2x = sin2x

Also Read: Simplify 1-sin²x | Formula of 1+tan²x

Note that if we substitute θ in place of x, we will get the formula of 1-cos²θ which is provided below:

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

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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 $1/2$.

More Trigonometric Identities:

sinx siny Formula | sinx cosy Formula

Formula of cot(x+y)

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