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

Question

If cos θ = \(\frac{12}{13}\), show that sin θ(1- tan θ) = \(\frac{35}{156}\)

Answer 1

Given: cos θ = \(\frac{12}{13}\)

To prove: sin θ (1 - tan θ) = \(\frac{35}{156}\)

Proof: we know,cos θ = \(\frac{B}{H}\)

Where B is base and H is hypotenuse of the right angled triangle.We construct a right triangle ABC right angled at B such that ∠ACB = θ

Perpendicular is AB, Base is BC = 12 and hypotenuse is AC = 13.In the triangle ABC,By Pythagoras theorem, we have

AC² = AB² + BC²

13² = AB²+12²

169 = AB²+144

169 -144 = AB²

25= AB² 

AB = \(\sqrt{25}\) = 5

sin θ = \(\frac{P}{H}\) = \(\frac{5}{13}\)

so,

tan θ = \(\frac{P}{H}\) = \(\frac{5}{12}\)

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

sinθ(1 - tanθ) = \(\frac{5}{13}\) \((1-\frac{5}{12})\) = \(\frac{5}{13}\)x \(\frac{7}{12}\) = \(\frac{35}{156}\)

Hence Proved.