First slide

Algebra of real valued functions

Question

If $f (x) = \sin^{2} x + \sin^{2} (x + π / 3) + \cos x \cos (x + π / 3)$ and g(5/4) = 1 then $(g o f) (x) =$

Moderate

A

1
B

0
C

sinx
D

-cosx

Solution

$f (x) = \sin^{2} x + {(\sin x \cos \frac{π}{3} + \cos x \sin \frac{π}{3})}^{2} + \cos x$
$(\cos x \cos \frac{π}{3} - \sin x \sin \frac{π}{3})$
$= \sin^{2} x + [\frac{\sin x}{2} + \frac{\sqrt{3} \cos x}{2}] + \frac{\cos^{2} x}{2} - \frac{\sqrt{3}}{2} \cos x \sin x$
$- \sin^{2} x + \frac{\sin^{2} x}{4} + \frac{3}{4} \cos^{2} x + \frac{\sqrt{3}}{2}$
$\sin x \cos x + \frac{\cos^{2} x}{2} + \frac{\sqrt{3}}{2}$
$\cos x \sin x = \frac{5}{4} (\sin^{2} x + \cos^{2} x) = \frac{5}{4}$
$∴ [g o f] (x) = g [f (x)] = g (5 / 4) = 1$

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