Question

if 1+sin^2 theta=3 sin theta cos theta then prove that tan theta=1 or 1/2

Answer 1

1+sin^2 theta=3 sin theta cos theta (we know that sin^2 theta + cos^2 theta =1)

= ( sin^2 theta + cos^2 theta  ) + sin ^2 theta = 3 sin theta cos theta

= sin^2 theta + cos^2 theta + sin ^2 theta = 3 sin theta cos theta

= cos^2 theta + 2 sin^2 theta = 3 sin theta cos theta

On dividing by cos^2 theta, we get

= 1 + 2 tan^2 theta = 3 tan theta

Let tan theta = b

 2b^2  - 3b + 1 = 0

= (2b-1)(b-1) = 0

b = 1 or 1/2

So, tan theta = 1 or 1/2.


Hope this helps!

Answer 2

Answer:

Given: 1 + Sin²Ф = 3 SinФ. CosФ

Using the formula: Sin²Ф + Cos²Ф = 1, we get,

⇒ ( Sin²Ф + Cos²Ф ) + Sin²Ф = 3 SinФ. CosФ

⇒ Cos²Ф + 2Sin²Ф = 3 SinФ. CosФ

Dividing both sides by Cos²Ф, we get,

⇒ 1 + 2 ( Sin²Ф / Cos²Ф ) = 3 SinФ. Cos²Ф / CosФ

Cos²Ф / CosФ will become 1 / CosФ which on multiplied by SinФ becomes SinФ / CosФ which is TanФ.

⇒ 1 + 2 Tan²Ф = 3 TanФ

Transposing 3 TanФ this side we get,

⇒ 2 Tan²Ф - 3 TanФ + 1 = 0

For sake of simplicity, let us take TanФ as 'x'

⇒ 2x² - 3x + 1 = 0

Solving the equation we get,

⇒ 2x² - 2x - x + 1 = 0

⇒ 2x ( x - 1 ) -1 ( x - 1 ) = 0

⇒ ( 2x - 1 ) ( x - 1 ) = 0

⇒ 2x = 1 , x = 1 / 2

⇒ x = 1

Hence the values of x are ( 1/2, 1 )

Hence Tan Ф values are ( 1/2, 1 )

This is the required values.

Hence Proved !!