Find the value of (cos θ + sin θ)/(cos θ – sin θ).

If tan θ = 5/12, find the value of (cos θ + sin θ)/(cos θ – sin θ).

Answer :

Consider ∆ABC be right angled at B and ∠ACB = θ

tan θ = AB/BC = 5/12

Take AB = 5x then BC = 12x

In right angled ∆ABC,

AC² = AB² + BC²

AC² = (5x)² + (12x)²

AC² = 25x² + 144x² = 169x²

AC² = (13x)²

⇒ AC = 13x

In right angled ∆ABC

cos θ = base/hypotenuse

cos θ = BC/AC

cos θ = 12x/13x = 12/13

In right angled ∆ABC

sin θ = perpendicular/hypotenuse

⇒ sin θ = AB/AC

sin θ = 5x/13x = 5/13

(cos θ + sin θ)/(cos θ – sin θ) = [12/13 + 5/13]/ [12/13 – 5/13]

Taking LCM

= [(12 + 5)/13]/[(12 – 5)/13]

= (17/13)/(7/13)

= 17/13 × 13/7

= 17/7

Hence,

(cos θ + sin θ)/(cos θ – sin θ) = 17/7

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