# If 1 + sin^{2} θ = 3sinθ cosθ , then prove that tanθ = 1 or 1/2

**Solution:**

Given, 1 + sin^{2}θ = 3sinθ cosθ

We have to prove that tanθ = 1 or 1/2.

Dividing by sin^{2}θ on both sides,

1/sin^{2}θ + sin^{2}θ/sin²θ = 3sinθ cosθ/sin^{2}θ

1/sin^{2}θ + 1 = 3cosθ/sinθ

We know that cosec A = 1/sin A

Also, cos A/sin A = cot A

cosec^{2}θ + 1 = 3cotθ ------------ (1)

By using trigonometric identity,

cot^{2 }A + 1 = cosec^{2} A

cot^{2} θ + 1 + 1 = 3cotθ

cot^{2 }θ - 3cotθ + 2 = 0

Let cotθ = x

So, x^{2} - 3x + 2 = 0

On factoring,

x^{2 }- x - 2x + 2 = 0

x(x - 1) - 2(x - 1) = 0

(x - 1)(x - 2) = 0

Now, x - 1 = 0

x = 1

Also, x - 2 = 0

x = 2

Now, cot θ = 1, 2

We know that tan θ = 1/cotθ

So, tan θ = 1/1 or 1/2

Therefore, tan θ = 1 or 1/2.

**✦ Try This: **If cosθ = √3/2, prove that: 3sinθ - 4sin^{3}θ = 1.

**☛ Also Check:** NCERT Solutions for Class 10 Maths Chapter 8

**NCERT Exemplar Class 10 Maths Exercise 8.4 Problem 4**

## If 1 + sin^{2} θ = 3sinθ cosθ , then prove that tanθ = 1 or 1/2

**Summary:**

The sine function is written as the ratio of the length of the perpendicular and hypotenuse of the right-angled triangle. If 1 + sin² θ = 3sinθ cosθ, then it is proven that tanθ = 1 or 1/2

**☛ Related Questions:**

visual curriculum