The formula of cos^2x – sin^2x is equal to cos2x. Let us learn how to establish the proof the identity of cos^{2}x-sin^{2}x.

cos^{2}x-sin^{2}x formula is given by cos^{2}x-sin^{2}x = cos2x.

## Proof of cos^{2}x-sin^{2}x Formula

**Question:** Prove that cos^{2}x-sin^{2}x = cos2x.

*Proof:*

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

= cosx ⋅ cosx – sinx ⋅ sinx

= cos(x+x), here we have used the identity cosa cosb – sina sinb = cos(a+b).

= cos 2x

So cos^2x – sin^2x is equal to cos2x and we have proved this cos^{2}x-sin^{2}x formula by using the identity of cos(a+b).

## Question-Answer

**Question 1:** Find the value of cos^{2}45° -sin^{2}45°.

*Answer:*

Putting x=45° in the above formula cos^{2}x-sin^{2}x = cos2x, we obtain that

cos^{2}45° -sin^{2}45°

= cos (2×45)°

= cos90°

= 0 as the value of cos90° is 0.

Thus the value of cos^{2}45° -sin^{2}45° is equal to 0.

**Related Topics:**

## FAQs

**Q1: What is cos^2x – sin^2x?**

Answer: cos^{2}x-sin^{2}x is equal to cos2x.