Let's Review a Little More...
You also learned how to use trigonometric identities to simplify expressions. To do this, you applied techniques you learned from algebra such as factoring, common denominators, the distributive property, and other techniques. The basic steps are shown in the table below.
| Step 1 | Determine what identities can be used to rewrite the expression. |
| Step 2 | Rewrite and simplify the expression. |
| Step 3 | Repeat Steps 1 and 2 as many times as needed to simplify the expression. |
Review simplifying trigonometric expressions using trigonometric identities by completing the activity below. Click each tab and simplify each expression completely, then check your answer.
What is the simplified expression of \( \csc x - \csc x \cos^{2} x \)?
The answer is \( \sin x \).
If you need help arriving at this answer, click the Solution button.
Factor \( \csc x \) from both terms. |
\( \csc x - \csc x \cos^{2} x = \csc x(1 - \cos^{2} x) \) |
Determine what identities can be used to rewrite the expression. |
Use the cosecant reciprocal identity, \( \csc\theta = \frac{1}{\sin\theta} \). \( \csc x = \frac{1}{\sin x} \) |
Rewrite and simplify the expression. |
Substitute the identity into the expression to give: \( \csc x \left( 1 - \cos^{2} x \right) = \frac{1}{\sin x}(1 - \cos^{2} x) \) |
Determine what identities can be used to rewrite the expression. |
Use the Pythagorean identity with sines and cosines, \( \sin^{2}\theta + \cos^{2}\theta = 1 \) \( \sin^{2} x + \cos^{2} x = 1 \) \( \sin^{2} x = 1 - \cos^{2} x \) |
Rewrite and simplify the expression. |
Substitute the identity into the expression to give: \( \frac{1}{\sin x}\left( 1 - \cos^{2} x \right) = \frac{1}{\sin x}(\sin^{2} x) \) Simplify the expression: \( \frac{1}{\sin x}\left( \sin^{2} x \right) = \sin x \) The final simplified expression is \( \sin x \). |
What is the simplified expression of \( \frac{\sec x - \sin x \tan x}{\cos x} \)?
The answer is 1.
If you need help arriving at this answer, click the Solution button.
Determine what identities can be used to rewrite the expression. |
Use the tangent quotient identity, \( \tan\theta = \frac{\sin\theta}{\cos\theta} \). \( \tan x = \frac{\sin x}{\cos x} \). |
Rewrite and simplify the expression. |
Substitute the identity into the original expression to give: \( \frac{\sec x - \sin x \tan x}{\cos x} = \frac{\sec x - \sin x \frac{\sin x}{\cos x}}{\cos x} \) Simplify the numerator. \( \frac{\sec x - \frac{\sin^{2} x}{\cos x}}{\cos x} \) |
Determine what identities can be used to rewrite the expression. |
Use the secant quotient identity: \( \sec\theta = \frac{1}{\cos\theta} \). \( \sec x = \frac{1}{\cos x} \) |
Rewrite and simplify the expression. |
Substitute the identity into the expression to give: \( \frac{\sec x - \frac{\sin^{2} x}{\cos x}}{\cos x} = \frac{\frac{1}{\cos x} - \frac{\sin^{2} x}{\cos x}}{\cos x} \) Rewrite the numerator: \( \frac{\frac{1 - \sin^{2} x}{\cos x}}{\cos x} \) |
Determine what identities can be used to simplify the expression. |
Use the Pythagorean identity with sines and cosines, \( \sin^{2}\theta + \cos^{2}\theta = 1 \) \( \sin^{2} x + \cos^{2} x = 1 \) \( \cos^{2} x = 1 - \sin^{2} x \) |
Rewrite and simplify the expression. |
Substitute the identity into the expression to give: \( \frac{\frac{\cos^{2} x}{\cos x}}{\cos x} \) Simplify the expression: \( \frac{\frac{\cos^{2} x}{\cos x}}{\cos x} = \frac{\cos x}{\cos x} = 1 \) The final simplified expression is 1. |
