The principal value of ^{-1}≤ft[{1}{{2}}≤ft( 9 π/10- 9 π/10)] is

Correct Answer is : 17π2017 π/202017π

Correct Answer is : 17π2017 π/202017π

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MHTCET 25 Apr 2025 Shift 1 Paper

Question : 117 of 150

Marks: +1, -0

\cos ^{-1}\left[\frac{1}{\sqrt{2}}\left(\cos \frac{9 \pi}{10}-\sin \frac{9 \pi}{10}\right)\right]

Solution:

Concept:

The principal value of

\cos^{-1}x

always lies in the interval

[0,\pi]

If the angle inside

\cos^{-1}

is outside this range, we convert it using cosine properties.

Explanation:

Consider

E = \frac{1}{\sqrt{2}}\left(\cos\frac{9\pi}{10} - \sin\frac{9\pi}{10}\right)

Use the identity

\frac{1}{\sqrt{2}}(\cos A - \sin A) = \cos\left(A+\frac{\pi}{4}\right)

So,

E = \cos\left(\frac{9\pi}{10} + \frac{\pi}{4}\right) = \cos\frac{23\pi}{20}

Therefore, the expression becomes

\cos^{-1}\left[\cos\frac{23\pi}{20}\right]

Since

\frac{23\pi}{20} > \pi

, it is not in the principal value range

[0,\pi]

Using

\cos(2\pi - \theta) = \cos\theta

, we get

\cos\frac{23\pi}{20} = \cos\frac{17\pi}{20}

Now

\frac{17\pi}{20}

lies in

[0,\pi]

, so it is the required principal value.

Answer:

\frac{17\pi}{20}

Option B.

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