Question

If tan(πcosθ) = cot(πsinθ), then the value of cos(θ-π/4) is (are)

• A. 1/2
• B. 1/$\sqrt { 2 }$
• C. $\pm \frac { 1 } { 2 \sqrt { 2 } }$
• D. None of these

Answer 1

Answer: (C)

$\pm \frac { 1 } { 2 \sqrt { 2 } }$

Answer 2

We have tan(πcosθ) = cot(πsinθ) ⇒ tan(πcosθ)

= tan(π/2 - πsinθ)

⇒ π cosθ = (π/2 - πsinθ) + nπ , n ∈ z

⇒ cosθ + sinθ = $\frac { 1 } { 2 }$ + n , n ∈ z

⇒ $\frac { 1 } { \sqrt { 2 } }$cosθ +$\frac { 1 } { \sqrt { 2 } }$ sinθ = , n ∈ z

⇒ cos$\left( \theta - \frac { \pi } { 4 } \right)$=, n ∈ z

⇒ cos$\left( \theta - \frac { \pi } { 4 } \right)$= ± $\frac { 1 } { 2 \sqrt { 2 } }$ ( for n =0 and n = -1)