Chapter 5: Q 10. (page 495)

Pythagorean identities: Use Pythagorean identities to rewrite each of the following trigonometric expressions.
$\csc^{8} x =____$

Short Answer

Expert verified

$\frac{1}{8} [1 - \cos 2 x \frac{(\cos 4 x + 1)}{2}]$

Step by step solution

Step 1. Given information.

Consider the given information,

$\csc^{8} x$

Step 2. Calculation.

Simplify the expression with the Pythagorean theorem.

$\begin{array}{rcl} \cos^{6} x & = & {(\cos^{2} x)}^{3} \\ = & {(1 - \sin^{2} x)}^{3} \\ = & {(1 - \frac{1 - \cos 2 x}{2})}^{3} \\ = & \frac{1}{8} {(1 - \cos 2 x)}^{3} \\ = & \frac{1}{8} (1 - \cos 2 x) (1 + \cos 2 x + \cos^{2} 2 x) \\ = & \frac{1}{8} [1 + \cos 2 x + \cos^{2} 2 x - \cos 2 x (1 + \cos 2 x + \cos^{2} 2 x)] \\ = & \frac{1}{8} [1 + \cos 2 x + \frac{\cos 4 x + 1}{2} - (\cos 2 x + \frac{\cos 4 x + 1}{2} + \cos 2 x \frac{\cos 4 x + 1}{2})] \\ = & \frac{1}{8} [1 + \cos 2 x + \frac{(\cos 4 x + 1)}{2} - \cos 2 x - \frac{(\cos 4 x + 1)}{2} - \cos 2 x \frac{(\cos 4 x + 1)}{2}] \\ = & \frac{1}{8} [1 - \cos 2 x \frac{(\cos 4 x + 1)}{2}] \end{array}$

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Pythagorean identities: Use Pythagorean identities to rewrite each of the following trigonometric expressions.
$\csc^{8} x =____$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Calculation.

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Pythagorean identities: Use Pythagorean identities to rewrite each of the following trigonometric expressions.csc8x=____

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Calculation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Geometry

Discrete Mathematics

Decision Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Pythagorean identities: Use Pythagorean identities to rewrite each of the following trigonometric expressions.
$\csc^{8} x =____$