What is {sinθ -2sin³θ }{2cos³θ -cosθ } equal to?

Correct Answer is : tanθtanθtanθ

Correct Answer is : tanθtanθtanθ

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UPSC CDS II 2024 Math Paper

Question : 40 of 100

Marks: +1, -0

What is

\frac{sin\theta -2sin^{3}\theta }{2cos^{3}\theta -cos\theta }

equal to?

Solution:

Concept:

Simplify the trigonometric fraction by factoring and using the identity

\sin^2 \theta + \cos^2 \theta = 1

Explanation:

Start with the given expression:

\frac{\sin\theta - 2\sin^3\theta}{2\cos^3\theta - \cos\theta}

Factor the numerator:

\sin\theta - 2\sin^3\theta = \sin\theta(1 - 2\sin^2\theta)

Factor the denominator:

2\cos^3\theta - \cos\theta = \cos\theta(2\cos^2\theta - 1)

Rewrite

1 - 2\sin^2\theta

using

\sin^2\theta = 1 - \cos^2\theta

1 - 2(1 - \cos^2\theta) = 1 - 2 + 2\cos^2\theta = 2\cos^2\theta - 1

Thus the numerator becomes

\sin\theta(2\cos^2\theta - 1)

Now the expression is

\frac{\sin\theta(2\cos^2\theta - 1)}{\cos\theta(2\cos^2\theta - 1)}

Cancel the common factor

(2\cos^2\theta - 1)

(provided

\cos^2\theta \neq \frac{1}{2}

This simplifies to

\frac{\sin\theta}{\cos\theta} = \tan\theta

Answer:

\tan\theta

Hence the correct option is D.

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