Question

If 4 tan θ = 3 then prove that sin θ cos θ = `12/25`.

Answer 1

Given: 4 tan θ = 3.

To Prove: sin θ cos θ = `12/25`

Proof [Step-wise]:

1. From 4 tan θ = 3, tan θ = `3/4`.

2. Write sin θ and cos θ in the same ratio:

Let sin θ = 3k and cos θ = 4k for some real k.   ...`("Since" tan θ = sin θ/cos θ = 3/4)`

3. Use the Pythagorean identity:

sin²θ + cos²θ = 1

⇒ (3k)² + (4k)² = 1

⇒ 9k² + 16k² = 25k² = 1

⇒ `k^2 = 1/25`

4. Compute sin θ cos θ = (3k)(4k)

= 12k² 

= `12 xx 1/25` 

= `12/25`

`sin θ cos θ = 12/25`, as required.