If sin 4 x 2 + cos 4 x 3 = 1 5 , then [2009]

Question

If $\frac{\sin^{4} x}{2} + \frac{\cos^{4} x}{3} = \frac{1}{5}$ , then [2009]

Smart Achievers · Accepted Answer

(1, 2)

Given:

$\frac{\sin^{4} x}{2} + \frac{\cos^{4} x}{3} = \frac{1}{5} \Rightarrow 3 \sin^{4} x + 2 \cos^{4} x = \frac{6}{5}$

$\Rightarrow \sin^{4} x + 2 [\sin^{4} x + \cos^{4} x] = \frac{6}{5}$

$\Rightarrow \sin^{4} x + 2 [1 - 2 \sin^{2} x \cos^{2} x] = \frac{6}{5}$

$\Rightarrow \sin^{4} x + 2 - 4 \sin^{2} x (1 - \sin^{2} x) = \frac{6}{5}$

$\Rightarrow 5 \sin^{4} x - 4 \sin^{2} x + 2 - \frac{6}{5} = 0$

$\Rightarrow 25 \sin^{4} x - 20 \sin^{2} x + 4 = 0$

$\Rightarrow {(5 \sin^{2} x - 2)}^{2} = 0 \Rightarrow \sin^{2} x = \frac{2}{5}$

$\Rightarrow \cos^{2} x = \frac{3}{5} and \tan^{2} x = \frac{2}{3}$

$Also \frac{\sin^{8} x}{8} + \frac{\cos^{8} x}{27} = \frac{2}{625} + \frac{3}{625} = \frac{5}{625} = \frac{1}{125}$

Want Us to Email you About Special Offers & Updates?