Questions: Find the exact value of the expression. Do not use a calculator tan (π 5)/4 + tan (5 π)/4 (23+1)/2 (2+2+1)/6 sqrt(3)/2 0 None of these 1/2

Find the exact value of the expression. Do not use a calculator tan (π 5)/4 + tan (5 π)/4 (23+1)/2 (2+2+1)/6 sqrt(3)/2 0 None of these 1/2
Transcript text: Find the exact value of the expression. Do not use a calculator \[ \tan \frac{\pi 5}{4}+\tan \frac{5 \pi}{4} \] $\frac{23+1}{2}$ $\frac{2+2+1}{6}$ $\frac{\sqrt{3}}{2}$ 0 None of these $\frac{1}{2}$
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Solution

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Solution Steps

Step 1: Evaluate the Angle

We start by evaluating the angle \( \frac{5\pi}{4} \). This angle is located in the third quadrant of the unit circle.

Step 2: Calculate the Tangent

Next, we calculate \( \tan\left(\frac{5\pi}{4}\right) \). In the third quadrant, the tangent function is positive, and we find that: \[ \tan\left(\frac{5\pi}{4}\right) = 1 \]

Step 3: Sum the Tangent Values

Since the expression requires us to find \( \tan\left(\frac{5\pi}{4}\right) + \tan\left(\frac{5\pi}{4}\right) \), we can sum the two values: \[ \tan\left(\frac{5\pi}{4}\right) + \tan\left(\frac{5\pi}{4}\right) = 1 + 1 = 2 \]

Final Answer

\(\boxed{2}\)

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