Question

If `cos θ = 12/13`, show that `sin θ (1 - tan θ) = 35/156`.

If `cos θ = 12/13`, verify that `sin θ (1 - tan θ) = 35/156`.

Answer 1

Given: cos θ = `12/13`

To prove: sin θ (1 – tan θ) = `35/156`

Proof: We know, cos θ = `B/H`

where the right-angled triangle’s base is B and its hypotenuse is H. ∠ACB = 8 is achieved by building a right triangle ABC at a right angle to B. 

AB is perpendicular, BC = 12 is base, and AC = 13 is hypotenuse.

According to Pythagoras theorem, we have

AC² = AB² + BC²

13² = AB² + 12²

169 = AB² + 144

169 – 144 = AB²

25 = AB²

AB = `sqrt25`

AB = 5

sin θ = `P/H = 5/13`

So, tan θ = `P/H = 5/12`

Put the values in sin θ (1 – tan θ) to find its value,

sin θ (1 – tan θ) = `15/3 (1 - 5/12)`

= `5/13 xx 7/12`

= `35/156`

Hence Proved.