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Exercises Trigonometry

Example 6 from Algebra and Trigonometry, 7.4 The Other Trigonometric Functions

Using Identities to Simplify Trigonometric Expressions

Simplify $\frac{\sec t}{\tan t} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We can simplify this by rewriting both functions in terms of sine and cosine.

$\begin{matrix} \frac{sec t}{tan t} & = & \frac{\frac{1}{cos t}}{\frac{sin t}{cos t}} \\ = & \frac{1}{cos t} \cdot \frac{cos t}{sin t} & Multiply by the reciprocal . \\ = & \frac{1}{sin t} = csc t & Simplify and use the identity . \end{matrix}$
By showing that $\frac{\sec t}{\tan t}$ can be simplified to $\csc t,$ we have, in fact, established a new identity.
$\frac{\sec t}{\tan t} = \csc t$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Simplify $(\tan t) (\cos t) .$

How did it go?

PreviousUsing Identities to Evaluate Trigonometric Functions Next Using Identities to Relate Trigonometric Functions

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 7.4, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.