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JEE Main · 2023 · 08 Apr 2023 · Shift 2 · Trigonometric Ratios and Identities

The value of

36 (4 cos^{2} 9^{\circ} - 1) (4 cos^{2} 2 7^{\circ} - 1) (4 cos^{2} 8 1^{\circ} - 1) (4 cos^{2} 24 3^{\circ} - 1)

is

(A)54
(B)18
(C)27
(D)36✓ Correct

Answer: (D)

Solution

Using the identity

4 cos^{2} θ - 1 = \frac{s i n 3 θ}{s i n θ}

:

36 \cdot \frac{sin 2 7 ^{\circ}}{sin 9 ^{\circ}} \cdot \frac{sin 8 1 ^{\circ}}{sin 2 7 ^{\circ}} \cdot \frac{sin 24 3 ^{\circ}}{sin 8 1 ^{\circ}} \cdot \frac{sin 72 9 ^{\circ}}{sin 24 3 ^{\circ}}

This telescopes to:

36 \cdot \frac{sin 72 9 ^{\circ}}{sin 9 ^{\circ}}

Since

72 9^{\circ} = 2 \times 36 0^{\circ} + 9^{\circ}

, we have

sin 72 9^{\circ} = sin 9^{\circ}

.

Therefore:

36 \cdot \frac{sin 9 ^{\circ}}{sin 9 ^{\circ}} = 36

JEE Main · 2023 · 08 Apr 2023 · Shift 2 · Single correct option

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