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COMEDK 2025 Morning Solved Paper

Question : 154 of 180

Marks: +1, -0

The value of

\frac{1}{2 \sin 10^{\circ}}-2 \sin 70^{\circ}

Solution:

Concept:

Use trigonometric identities and the triple-angle formula to simplify the given expression.

Explanation:

Let

s = \sin 10^\circ

The expression becomes

E = \frac{1}{2\sin 10^\circ} - 2\sin 70^\circ = \frac{1}{2s} - 2\cos 20^\circ

(since

\sin 70^\circ = \cos 20^\circ

Use

\cos 20^\circ = 1 - 2\sin^2 10^\circ = 1 - 2s^2

Then

E = \frac{1}{2s} - 2(1 - 2s^2) = \frac{1}{2s} - 2 + 4s^2

Combine into a single fraction:

E = \frac{1 - 4s + 8s^3}{2s}

Apply the triple-angle identity for

\theta = 10^\circ

\sin 30^\circ = 3\sin 10^\circ - 4\sin^3 10^\circ

Thus

\frac{1}{2} = 3s - 4s^3

, which gives

8s^3 - 6s + 1 = 0

The numerator

1 - 4s + 8s^3 = (8s^3 - 6s + 1) + 2s = 0 + 2s = 2s

Therefore

E = \frac{2s}{2s} = 1

Answer:

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