Let fk (x) = \frac{1}{k} \left(\right. \left(sin\right)^{k} x + \left(cos\right)^{k} x \left.\right) where x \in R and k \geq 1. Then f4(x)

Engineering

Mathematics

Trigonometric Ratio of Multiple and Submultiples Angles

Question

Let f_k (x) = $\frac{1}{k} (\sin^{k} x + \cos^{k} x)$ where x $\in$ R and k $\geq$ 1. Then f₄(x) – f₆(x) equals

$\frac{1}{12}$

$\frac{1}{4}$

$\frac{1}{3}$

$\frac{1}{6}$

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f₄(x) – f₆(x)

$= \frac{1}{4} (1 - \frac{\sin^{2} 2 x}{2}) - \frac{1}{6} (1 - \frac{3}{4} \sin^{2} 2 x)$

$= \frac{1}{4} (1 - \frac{\sin^{2} 2 x}{2}) - \frac{1}{6} (1 - \frac{3}{4} \sin^{2} 2 x) = \frac{1}{12}$