Problem 32
Question
Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$\frac{\tan \frac{\pi}{5}+\tan \frac{4 \pi}{5}}{1-\tan \frac{\pi}{5} \tan \frac{4 \pi}{5}}$$
Step-by-Step Solution
Verified Answer
The expression reduces to \(\tan(\pi)\) which has a value of 0.
1Step 1: Identify formula
Here we identify that this is a tangent sum formula: \(\tan(a+b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}\).
2Step 2: Apply formula
By applying the formula, we get: \(\tan\left(\frac{\pi}{5}+\frac{4\pi}{5}\right) = \frac{\tan \frac{\pi}{5}+\tan \frac{4 \pi}{5}}{1-\tan \frac{\pi}{5} \tan \frac{4 \pi}{5}}\). The angle now is \(\pi\).
3Step 3: Evaluate the expression
Next, we substitute \(\pi\) into \(\tan(\pi)\) which gives us 0.
Other exercises in this chapter
Problem 32
In Exercises \(23-34\), verify each identity. $$\sin 2 t-\cot t=-\cot t \cos 2 t$$
View solution Problem 32
Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$\tan \frac{x}{2}=\frac{\sqrt{3}}{3}$$
View solution Problem 33
Verify each identity. $$\sec ^{2} x \csc ^{2} x=\sec ^{2} x+\csc ^{2} x$$
View solution Problem 33
In Exercises \(23-34\), verify each identity. $$\sin 4 t=4 \sin t \cos ^{3} t-4 \sin ^{3} t \cos t$$
View solution