# How do you use the angle sum or difference identity to find the exact value of #tan((7pi)/12)#?

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To find the exact value of (\tan\left(\frac{7\pi}{12}\right)) using the angle sum or difference identity, we can express (\frac{7\pi}{12}) as the difference between two angles whose tangent values are known.

(\frac{7\pi}{12}) can be expressed as (\frac{\pi}{3} - \frac{\pi}{4}).

Since (\tan(\alpha - \beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}), we can use this formula to find (\tan\left(\frac{7\pi}{12}\right)) by substituting (\alpha = \frac{\pi}{3}) and (\beta = \frac{\pi}{4}):

[\tan\left(\frac{7\pi}{12}\right) = \tan\left(\frac{\pi}{3} - \frac{\pi}{4}\right) = \frac{\tan\left(\frac{\pi}{3}\right) - \tan\left(\frac{\pi}{4}\right)}{1 + \tan\left(\frac{\pi}{3}\right)\tan\left(\frac{\pi}{4}\right)}]

Knowing that (\tan\left(\frac{\pi}{3}\right) = \sqrt{3}) and (\tan\left(\frac{\pi}{4}\right) = 1), we can substitute these values into the formula:

[\tan\left(\frac{7\pi}{12}\right) = \frac{\sqrt{3} - 1}{1 + \sqrt{3} \times 1}]

Simplify the expression:

[\tan\left(\frac{7\pi}{12}\right) = \frac{\sqrt{3} - 1}{1 + \sqrt{3}}]

To rationalize the denominator, multiply both numerator and denominator by the conjugate of the denominator:

[\tan\left(\frac{7\pi}{12}\right) = \frac{(\sqrt{3} - 1)(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})}]

[= \frac{-2\sqrt{3} + 1 + \sqrt{3} - 1}{1 - 3}]

[= \frac{-\sqrt{3}}{-2}]

Finally, simplify:

[\tan\left(\frac{7\pi}{12}\right) = \frac{\sqrt{3}}{2}]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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