Question

Use Pythagorean identities to rewrite each of the following trigonometric expressions.

${\sec }^{6}x=\_\_\_\_\_$

Short answer

The obtained result is$\frac{1}{{\left(1-{\sin }^{2}x\right)}^3}$.

Step-by-step solution

Step 1. Given information.

Consider the given expression.

${\sec }^{6}x$

Step 2. Apply the Pythagorean identities to rewrite the expression.

The Pythagorean identity is defined as,

${\sin }^{2}x+{\cos }^{2}x=1$

Use the identity to write the given expression.

$\begin{array}{rcl}{\cos }^{2}x& =& 1-{\sin }^{2}x\\ {\left({\cos }^{2}x\right)}^3& =& {\left(1-{\sin }^{2}x\right)}^3\\ {\cos }^{6}x& =& {\left(1-{\sin }^{2}x\right)}^3\\ \frac{1}{{\cos }^{6}x}& =& \frac{1}{{\left(1-{\sin }^{2}x\right)}^3}\\ {\sec }^{6}x& =& \frac{1}{{\left(1-{\sin }^{2}x\right)}^3}\end{array}$