If A = sin 2 x + cos 4 x , then for all real x

Question

If $A = \sin^{2} x + \cos^{4} x,$ then for all real $x$

Smart Achievers · Accepted Answer

(4)

$A = \sin^{2} x + \cos^{4} x$

$\Rightarrow$ $A = 1 - \cos^{2} x + \cos^{4} x$

$=$ $\cos^{4} x - \cos^{2} x + \frac{1}{4} + \frac{3}{4}$

$=$ ${(\cos^{2} x - \frac{1}{2})}^{2} + \frac{3}{4}$ $\dots (i)$

where, $0 \leq {(\cos^{2} x - \frac{1}{2})}^{2} \leq \frac{1}{4}$ $\dots (i i)$

$∴$ $\frac{3}{4} \leq A \leq 1$

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