First slide

Trigonometric Identities

Question

If $0 < x < \frac{π}{2}$ and $\sin^{n} x + \cos^{n} x > 1$ then

Moderate

A

$n \in (2, \infty)$
B

$n \in (- \infty, 2)$
C

$n \in (- 1, 1)$
D

None of these

Solution

$\begin{array}{l} S i n c e 0 < x < \frac{π}{2} \\ ∴ 0 < \sin x < 1 a n d 0 < \cos x < 1 \\ W h e n n = 2, \sin^{n} x + \cos^{n} x = 1 w h e n n > 2, b o t h \\ \sin^{n} x a n d \cos^{n} x w i l l d e c r e a s e a n d h e n c e \sin^{n} x + \cos^{n} x < 1 \\ W h e n n < 2, b o t h \sin^{n} x a n d \cos^{n} x w i l l i n c r e a s e a n d \\ h e n c e \sin^{n} x + \cos^{n} x \geq 1 f o r n≤2 \\ T h u s \sin^{n} x + \cos^{n} x \geq 1 f o r n \leq 2 \end{array}$

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