Question

Use an identity to write each expression as a single trigonometric function or as a single number in exact form.

Do not use a calculator.

$\frac{\tan \left(34°\right)}{2(1-{\tan }^2\left(34°\right))}$

Answer 1

Explanation for the correct answer:

Step-1: Given data.

Given expression is $\frac{\tan \left(34°\right)}{2(1-{\tan }^2\left(34°\right))}$

Step-2: To solve the given expression into a single trigonometric function.

Consider the denominator of the function

$$\begin{gathered} \Rightarrow 2(1-{\tan }^2\left(34°\right)) \\ \Rightarrow 2\left({\sec }^2\left(34°\right)\right) \end{gathered}$$ ( Since ${\sec }^2\left(\theta \right)+{\tan }^2\left(\theta \right)=1$)

Consider the given expression now.

$$\begin{gathered} \Rightarrow \frac{\tan \left(34°\right)}{2\left({\sec }^2\left(34°\right)\right)} \\ \Rightarrow \frac{{\displaystyle \frac{\sin \left(34°\right)}{\cos \left(34°\right)}}}{2{\sec }^2\left(34°\right)} \\ \Rightarrow \frac{\sin \left(34°\right)\times {\cos }^2\left(34°\right)}{2\cos \left(34°\right)} \\ \Rightarrow \frac{1}{2}\sin \left(34°\right)\cos \left(34°\right) \\ \Rightarrow \frac{1}{4}\sin \left(2\times 34°\right)=\frac{1}{4}\sin \left(68°\right) \end{gathered}$$

Therefore, The given expression can be written as a single trigonometric expression $=\frac{1}{4}\sin \left(68°\right)$.