What is $(pq + q r + r p)$ equal to?

Question

Question

Consider the following for the three (03) items that follow: Let

p = sin 3 5^{\circ}

,

q = sin 2 5^{\circ}

and

r = sin (- 9 5^{\circ})

.

This question was previously asked in

NDA-I (Mathematics) Official Paper (Held On: 13 Apr, 2025)

Answer 1

Calculation:

Given,

The expression to simplify is pq + qr + rp, where:

$p = \sin(35^\circ) $

$q = \sin(25^\circ) $

$r = \sin(-95^\circ) = -\cos(5^\circ) $

Thus, we need to evaluate:

$ pq + qr + rp = \sin(35^\circ) \sin(25^\circ) + \sin(25^\circ) (-\cos(5^\circ)) + (-\cos(5^\circ)) \sin(35^\circ) $

Using trigonometric identities, we simplify the expression:

$ pq + qr + rp = -\sin^2(25^\circ) - \sin(95^\circ) \sin(35^\circ) $

Using the sum-to-product identity and known values for sine and cosine:

$ = \frac{1}{2} [-2 \sin(25^\circ) - 2 \sin(95^\circ) \cdot \sin(35^\circ)] $

$ = \frac{1}{2} [-1 + \cos(50^\circ) + \cos(130^\circ) - \cos(60^\circ)] $

$ = \frac{1}{2} [-1 + \cos(50^\circ) - \cos(50^\circ) - \frac{1}{2}] $

$ = \frac{1}{2} \left[ -1 - \frac{1}{2} \right] = -\frac{3}{4} $

∴ The value of pq + qr + rp is $ -\frac{3}{4} .$

Hence, the correct answer is Option 1.

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