First slide

Trigonometric equations

Question

The solution set of equations $\cos^{5} x = 1 + \sin^{4} x$ is

Easy

A

$\{2 n π, n \in Ι\}$
B

$\{\frac{n π}{2}, n \in Ι\}$
C

$\{4 n π, n \in Ι\}$
D

$\{n π, n \in Ι\}$

Solution

$\cos^{5} x = 1 + \sin^{4} x$
Minimum value of R.H.S is 1
and maximum value of L.H.S is 1.
so equality holds only when both sides are simultaneously equal and equal to 1.
$\begin{array}{l} \Rightarrow \cos x = 1 a n d \sin x = 0 \\ ⇒ \cos x = 1 \Rightarrow x = 2 n π \\ a n d \sin x = 0 \Rightarrow x = n π \\ S o, s o l u t i o n s e t is x = 2 n π, n \in I \end{array}$

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