If 1 + sin² θ = 3sinθ cosθ , then prove that tanθ = 1 or 1/2

Solution:

Given, 1 + sin²θ = 3sinθ cosθ

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

Dividing by sin²θ on both sides,

1/sin²θ + sin²θ/sin²θ = 3sinθ cosθ/sin²θ

1/sin²θ + 1 = 3cosθ/sinθ

We know that cosec A = 1/sin A

Also, cos A/sin A = cot A

cosec²θ + 1 = 3cotθ ------------ (1)

By using trigonometric identity,

cot² A + 1 = cosec² A

cot² θ + 1 + 1 = 3cotθ

cot² θ - 3cotθ + 2 = 0

Let cotθ = x

So, x² - 3x + 2 = 0

On factoring,

x² - 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 - 4sin³θ = 1.

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

---

NCERT Exemplar Class 10 Maths Exercise 8.4 Problem 4

If 1 + sin² 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:

• Given that sinθ + 2cosθ = 1, then prove that 2sinθ – cosθ = 2
• The angle of elevation of the top of a tower from two points distant s and t from its foot are compl . . . .
• The shadow of a tower standing on a level plane is found to be 50 m longer when Sun’s elevation is 3 . . . .