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Question: If tan (π cos θ) = cot (π sin θ), then sin (θ+ π/4) equals...

If tan (π cos θ) = cot (π sin θ), then sin (θ+ π/4) equals

A

12\frac{1}{\sqrt{2}}

B

12\frac{1}{2}

C

122\frac{1}{2\sqrt{2}}

D

32\frac{\sqrt{3}}{2}

Answer

122\frac{1}{2\sqrt{2}}

Explanation

Solution

  • Concept: Use the identity cot x = tan (π/2 - x).

  • Solution: Given tan (π cos θ) = cot (π sin θ).
    We can write cot (π sin θ) as tan (π/2 - π sin θ).
    So, tan (π cos θ) = tan (π/2 - π sin θ).
    This implies π cos θ = nπ + (π/2 - π sin θ) for some integer n.
    Dividing by π, we get cos θ = n + 1/2 - sin θ.
    Rearranging, cos θ + sin θ = n + 1/2.
    We know that cos θ + sin θ = √2 (sin θ cos(π/4) + cos θ sin(π/4)) = √2 sin (θ + π/4).
    So, √2 sin (θ + π/4) = n + 1/2.
    We know that the range of cos θ + sin θ is [-√2, √2].
    Therefore, -√2 ≤ n + 1/2 ≤ √2.
    Approximately, -1.414 ≤ n + 0.5 ≤ 1.414.
    Subtracting 0.5 from all parts: -1.914 ≤ n ≤ 0.914.
    The possible integer values for n are 0 and -1.

    Case 1: n = 0
    √2 sin (θ + π/4) = 0 + 1/2
    √2 sin (θ + π/4) = 1/2
    sin (θ + π/4) = 1 / (2√2)

    Case 2: n = -1
    √2 sin (θ + π/4) = -1 + 1/2
    √2 sin (θ + π/4) = -1/2
    sin (θ + π/4) = -1 / (2√2)

    Since the options are positive, we choose the positive value.

  • Final Answer: sin (θ + π/4) = 1 / (2√2)