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Virtual Math Learning Center Texas A&M University Virtual Math Learning Center
tags:
Fundamental Trigonometric Identity Trigonometry Trigonometry Series

Trig Identities Exercise 1

Author: Hannah Solomon

The following problem is solved in this video. It is recommended that you try to solve the problem before watching the video. You can click "Reveal Answer" to see the answer to the problem.

Problem: Use Reciprocal and Ratio Identities to simplify  
\[
\frac{\sin^2{x}}{\cos{x}} \cdot\csc{x}
\]

Answer: \(\tan x\)

Solution Method:The Reciprocal and Ratio Identities are the most basic identities. The three reciprocal identities simply define the reciprocal functions. Then the ratio identities define tangent and cotangent in terms of sine and cosine. 
 
Reciprocal and Ratio Identities
\begin{array}{cc}
 \csc{x}=\dfrac{1}{\sin{x}} & \tan{x} = \dfrac{\sin{x}}{\cos{x}} \\[8pt]
 \sec{x}=\dfrac{1}{\cos{x}} &    \cot{x} = \dfrac{\cos{x}}{\sin{x}}\\[8pt]
   \cot{x}=\dfrac{1}{\tan{x}}
\end{array}
When simplifying an equation using trig identities, our goal is to get to a single function or number. A helpful trick is to get everything in terms of sine and cosine, because all trig functions can be defined by those two, as you see from these identities. 
\begin{equation*}
    \frac{\sin^2{x}}{\cos{x}} \cdot \csc{x} =
    \frac{\sin^2{x}}{\cos{x}} \cdot \frac{1}{\sin{x}} =
    \frac{\sin{x}}{\cos{x}} =
    \tan{x}
\end{equation*}

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