Question

use a sum or difference identity to find the exact value of the expression
\tan \frac { pi } { 12 }
\tan \frac { pi } { 12 } = square
(simplify your answer, including any radicals. use integers or fractions for any number

Explanation:

Step1: Express \(\frac{\pi}{12}\) as a difference

We know that \(\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}\).
The tangent - difference identity is \(\tan(A - B)=\frac{\tan A-\tan B}{1 + \tan A\tan B}\). Here \(A=\frac{\pi}{3}\) and \(B = \frac{\pi}{4}\).

Step2: Substitute the values of \(\tan A\) and \(\tan B\)

We know that \(\tan\frac{\pi}{3}=\sqrt{3}\) and \(\tan\frac{\pi}{4}=1\).
Substitute these values into the formula \(\tan(A - B)=\frac{\tan A-\tan B}{1+\tan A\tan B}\), we get \(\tan(\frac{\pi}{3}-\frac{\pi}{4})=\frac{\tan\frac{\pi}{3}-\tan\frac{\pi}{4}}{1+\tan\frac{\pi}{3}\tan\frac{\pi}{4}}\).

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LATEXBLOCK0
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Step3: Rationalize the denominator

Multiply the numerator and denominator by the conjugate of the denominator \((\sqrt{3}-1)\)

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Answer:

\(2-\sqrt{3}\)