SATHEE: Trigonometrical Ratios and Identities 1 Question 20

Trigonometrical Ratios and Identities 1 Question 20

20. If $\frac{\sin^{4} x}{2} + \frac{\cos^{4} x}{3} = \frac{1}{5}$ , then

(2009)

(a) $\tan^{2} x = \frac{2}{3}$

(b) $\frac{\sin^{8} x}{8} + \frac{\cos^{8} x}{27} = \frac{1}{125}$

(d) $\frac{\sin^{8} x}{8} + \frac{\cos^{8} x}{27} = \frac{2}{125}$

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Solution:

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

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

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

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

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

$\begin{aligned} \Rightarrow & \sin^{2} x & = \frac{2}{5} \\ \cos^{2} x & = \frac{3}{5}, \tan^{2} x = \frac{2}{3} \\ ∴ & \frac{\sin^{8} x}{8} + \frac{\cos^{8} x}{27} & = \frac{1}{125} \end{aligned}$

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Trigonometrical Ratios and Identities 1 Question 20

20. If sin4⁡x2+cos4⁡x3=15, then

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20. If $\frac{\sin^{4} x}{2} + \frac{\cos^{4} x}{3} = \frac{1}{5}$ , then