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UPSC NDA I 2021 Mathematics Solved Paper

Question : 47 of 120

Marks: +1, -0

\text{The value of } x \text{, satisfying the equation } \log_{\cos x} \sin x = 1 \text{, where } 0

Solution:

Concept:

Use the change of base property of logarithms to remove the custom base and simplify the equation.

Explanation:

Start with

\log_{\cos x} \sin x = 1

Apply the change of base formula:

\frac{\log \sin x}{\log \cos x} = 1

Multiply both sides by

\log \cos x

to get

\log \sin x = \log \cos x

Since logarithms are one-to-one,

\sin x = \cos x

In the interval

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

, the smallest positive solution to

\sin x = \cos x

x = \frac{\pi}{4}

Answer:

\frac{\pi}{4}

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