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JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Trigonometric Equations

Let

S = {θ \in [0, 2 π) : tan (π cos θ) + tan (π sin θ) = 0}

.

Then

\sum_{θ \in S} sin^{2} (θ + \frac{π}{4})

is equal to

Answer: 2

Solution

The equation

tan (π cos θ) = - tan (π sin θ) = tan (- π sin θ)

requires

π cos θ = nπ - π sin θ

, that is

sin θ + cos θ = n

for an integer

n

. Since

∣ sin θ + cos θ ∣ \leq 2

, only

n = - 1, 0, 1

occur:

θ \in {0, \frac{π}{2}}

for

n = 1

,

θ \in {π, \frac{3 π}{2}}

for

n = - 1

, and

θ \in {\frac{3 π}{4}, \frac{7 π}{4}}

for

n = 0

. Using

sin^{2} (θ + \frac{π}{4}) = \frac{1 + sin 2 θ}{2}

, the six values give

\frac{1}{2}, \frac{1}{2}, 0, \frac{1}{2}, \frac{1}{2}, 0

. Their sum is

2

.

JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Integer answer

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