NDA 2023 Maths · Mathematics · Trigonometric Functions and Identities · hard

The minimum value of sin ^6x+cos ^6x is

Single correct option

The minimum value of

\sin^6x+\cos^6x

Show answer and solution

Answer: (C) $\dfrac14$

\sin^6x+\cos^6x=(\sin^2x+\cos^2x)^3-3\sin^2x\cos^2x=1-\frac34\sin^22x

. This is least when

\sin^22x=1

, giving

1-\frac34=\frac14

. The value

0

cannot occur because

\sin x

and

\cos x

never vanish together.

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