If α=1/7 and β={1}{{10}}, 0

Correct Answer is : π4−απ/4-α4π−α

Correct Answer is : π4−απ/4-α4π−α

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COMEDK 2025 Evening Solved Paper

Question : 178 of 180

Marks: +1, -0

\tan \alpha=\frac{1}{7}

\sin \beta=\frac{1}{\sqrt{10}}, \quad 0 < \alpha, \beta < \frac{\pi}{2}

2 \beta

is equal to

Solution:

Concept:

Use trigonometric identities to find

\tan(2\beta)

and compare it with

\tan(\frac{\pi}{4}-\alpha)

Explanation:

Given

\sin\beta = \frac{1}{\sqrt{10}}

and

0 < \beta < \frac{\pi}{2}

, compute

\cos\beta = \sqrt{1-\sin^2\beta} = \frac{3}{\sqrt{10}}

Thus

\tan\beta = \frac{\sin\beta}{\cos\beta} = \frac{1}{3}

Now find

\tan(2\beta)

using the double-angle formula:

\tan(2\beta) = \frac{2\tan\beta}{1-\tan^2\beta} = \frac{2\cdot\frac13}{1-\frac19} = \frac{\frac23}{\frac89} = \frac{3}{4}

Given

\tan\alpha = \frac17

, compute

\tan(\frac{\pi}{4}-\alpha)

using the difference formula:

\tan(\frac{\pi}{4}-\alpha) = \frac{1-\tan\alpha}{1+\tan\alpha} = \frac{1-\frac17}{1+\frac17} = \frac{\frac67}{\frac87} = \frac{3}{4}

Since

\tan(2\beta) = \tan(\frac{\pi}{4}-\alpha)

and both

2\beta

and

\frac{\pi}{4}-\alpha

lie in the interval

(0,\frac{\pi}{2})

(because

\alpha

and

\beta

are acute), the angles must be equal.

Therefore

2\beta = \frac{\pi}{4}-\alpha

Answer:

2\beta = \frac{\pi}{4}-\alpha

, which corresponds to Option B.

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