# What is Cos Square theta Formula?

The equations that relate the different trigonometric functions for any variable are known as** trigonometric identities**. These trigonometric identities help us to relate various trigonometric formulas and relationships with different angles. They are sine, cosine, tangent, cotangent, sec, and cosec. Here, we will look at the **cos square theta formula**.

According to the trigonometric identities, the cos square theta formula is given by

cos^{2}θ + sin^{2}θ = 1where θ is an acute angle of a right-angled triangle.

**Proof:**

The trigonometric functions for any right angled triangle is defined as:

cosθ = base/hypotenuse

sinθ = altitude/hypotenuseSo, we can write

cos

^{2}θ + sin^{2}θ = base^{2}/hypotenuse^{2}+ altitude^{2}/hypotenuse^{2}Thus, cos

^{2}θ + sin^{2}θ = (base^{2}+ altitude^{2})/hypotenuse^{2}Applying pyhogorus theorem for right angled triangle, we get

base

^{2}+ altitude^{2}= hypotenuse^{2}Thus, we get

cos^{2}θ + sin^{2}θ = 1

Other than this, there are some generalized formulas derived using this property:

- cos
^{2}θ = 1 – sin^{2}θ - cos2θ = cos
^{2}θ – sin^{2}θ - cos2θ = 2cos
^{2}θ – 1

### Sample Problems

**Question 1. Find the value of cosθ, given the value of sinθ, is 3/5.**

**Solution:**

Given, the value of sinθ = 3/5

Using cos square formula, we get

cos

^{2}θ + sin^{2}θ = 1cos

^{2}θ = 1 – sin^{2}θ = 1 – (3/5)^{2}= 1 – 9/25cos

^{2}θ = 16/25cosθ = √(16/25) = ± 4/5

Thus, the value of cosθ is

± 4/5.

**Question 2. Find the value of cosθ, given the value of cosθ – sinθ = 1**

**Solution:**

Given, cosθ – sinθ = 1.

or, cosθ = 1 + sinθ —- (i)

Using cos square formula, we get

cos

^{2}θ + sin^{2}θ = 1cos

^{2}θ = 1 – sin^{2}θ = (1 + sinθ)(1 – sinθ)cos

^{2}θ = cosθ (1 – sinθ)cosθ (cosθ – 1 + sinθ) = 0

So, we get two cases,

cosθ = 0else, cosθ – 1 + sinθ = 0

or, cosθ = 1 – sinθ —- (ii)

From eq.(i) and eq.(ii), we get

1 – sinθ = 1 + sinθ

2sinθ = 0

sinθ = 0

From eq.(i), we get cosθ = 1 + sinθ = 1 + 0 = 1

Thus,

cosθ = 1So, we get two possibilities. The value of cosθ is

0or1.

**Question 3. If cosθ = 3/5, find the value of sin ^{2}θ – cos^{2}θ.**

**Solution:**

Given, the value of cosθ = 3/5

Now, using cos square formula, we can write

sin

^{2}θ – cos^{2}θ = (1 – cos^{2}θ) – cos^{2}θ = 1 – 2cos^{2}θPutting the value of cosθ = 3/5, we get

sin

^{2}θ – cos^{2}θ = 1 – 2cos^{2}θ = 1 – 2 × (3/5)^{2}= 1 – 2 × 9/25 =1 – 18/25 =7/25So, the answer is

7/25.

**Question 4. Find the value of cos2θ, given the value of cosθ = 1/2.**

**Solution:**

Using the generalized formula,

cos2θ = 2cos^{2}θ – 1we can find the value of cos2θ, by substituting cosθ = 1/2

cos2θ = 2 × (1/2)

^{2}– 1 = 2/4 – 1 = 1/2 – 1 = – 1/2Thus,

cos2θ = – 1/2.