Correct Answer - Option 1 : 2/3
Given:
tanθ × tan2θ = 0.5
Concepts used:
tan2θ = 2tanθ/(1 – tan2θ)
tanθ = P/B
Calculation:
tanθ × 2tanθ/(1 – tan2θ) = ½
⇒ 4tan2θ/(1 – tan2θ) = 1
⇒ 4tan2θ = (1 – tan2θ)
⇒ tanθ = 1/√5
tanθ = P/B
On comparing,
P = 1, B = √5
Applying Pythagoras theorem,
H2 = P2 + B2
⇒ H2 = 12 + (√5)2
⇒ H = √6
cos2θ – sin2θ = (B/H)2 – (P/H)2
⇒ (√5/√6)2 – (1/√6)2
⇒ 5/6 – 1/6 = 4/6 = 2/3
∴ Value of cos2θ – sin2θ is 2/3.
