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Consider the following expression: (tan((pi)/(12))+tan((5pi)/(3)))/(1-tan((pi)/(12))tan((5pi)/(3))) Find the exact value of the expression without a calculator. [Hint: This diagram of special trigonometry values may help.] Choose 1 answer: (A) -1 (B) 1 (c) Undefined (D) 0

Consider the following expression:\newlinetan⁡(π12)+tan⁡(5π3)1−tan⁡(π12)tan⁡(5π3) \frac{\tan \left(\frac{\pi}{12}\right)+\tan \left(\frac{5 \pi}{3}\right)}{1-\tan \left(\frac{\pi}{12}\right) \tan \left(\frac{5 \pi}{3}\right)} \newlineFind the exact value of the expression without a calculator.\newline[Hint: This diagram ofspecial trigonometry values may help]\newlineChoose 11 answer:\newline(A) −1-1\newline(B) 11\newline(C) Undefined\newline(D) 00

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Full solution

Q. Consider the following expression:\newlinetan⁡(π12)+tan⁡(5π3)1−tan⁡(π12)tan⁡(5π3) \frac{\tan \left(\frac{\pi}{12}\right)+\tan \left(\frac{5 \pi}{3}\right)}{1-\tan \left(\frac{\pi}{12}\right) \tan \left(\frac{5 \pi}{3}\right)} \newlineFind the exact value of the expression without a calculator.\newline[Hint: This diagram ofspecial trigonometry values may help]\newlineChoose 11 answer:\newline(A) −1-1\newline(B) 11\newline(C) Undefined\newline(D) 00
  1. Simplify Tangent Values: First, let's simplify the individual tangent values using known trigonometric identities and special angles.\newlinetan⁡(π12)\tan\left(\frac{\pi}{12}\right) is not a standard angle, but tan⁡(5π3)\tan\left(\frac{5\pi}{3}\right) is equivalent to tan⁡(5π3−2π)=tan⁡(−π3)\tan\left(\frac{5\pi}{3} - 2\pi\right) = \tan\left(-\frac{\pi}{3}\right), which is a standard angle.
  2. Find tan⁡(π12)\tan(\frac{\pi}{12}): We know that tan⁡(−θ)=−tan⁡(θ)\tan(-\theta) = -\tan(\theta). So, tan⁡(−π3)=−tan⁡(π3)\tan(-\frac{\pi}{3}) = -\tan(\frac{\pi}{3}). The exact value of tan⁡(π3)\tan(\frac{\pi}{3}) is 3\sqrt{3}, so tan⁡(−π3)=−3\tan(-\frac{\pi}{3}) = -\sqrt{3}.
  3. Use Angle Subtraction Formula: Now, we need to find the value of tan⁡(π12)\tan\left(\frac{\pi}{12}\right). We can use the angle subtraction formula for tangent, tan⁡(a−b)=tan⁡(a)−tan⁡(b)1+tan⁡(a)tan⁡(b)\tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}, where we can let a=π4a = \frac{\pi}{4} and b=π6b = \frac{\pi}{6}, since tan⁡(π4)\tan\left(\frac{\pi}{4}\right) and tan⁡(π6)\tan\left(\frac{\pi}{6}\right) are known values.
  4. Plug Values into Expression: Using the angle subtraction formula, we get tan⁡(π12)=tan⁡(π4−π6)=tan⁡(π4)−tan⁡(π6)1+tan⁡(π4)tan⁡(π6)\tan\left(\frac{\pi}{12}\right) = \tan\left(\frac{\pi}{4} - \frac{\pi}{6}\right) = \frac{\tan\left(\frac{\pi}{4}\right) - \tan\left(\frac{\pi}{6}\right)}{1 + \tan\left(\frac{\pi}{4}\right)\tan\left(\frac{\pi}{6}\right)}.
  5. Simplify Numerator: The exact values for tan⁡(π4)\tan(\frac{\pi}{4}) and tan⁡(π6)\tan(\frac{\pi}{6}) are 11 and 3/3\sqrt{3}/3, respectively. Plugging these into the formula, we get tan⁡(π12)=1−3/31+(1)(3/3)\tan(\frac{\pi}{12}) = \frac{1 - \sqrt{3}/3}{1 + (1)(\sqrt{3}/3)}.
  6. Simplify Denominator: Simplify the expression for tan⁡(π12)\tan\left(\frac{\pi}{12}\right): tan⁡(π12)=3−33+3\tan\left(\frac{\pi}{12}\right) = \frac{3 - \sqrt{3}}{3 + \sqrt{3}}.
  7. Combine Denominator Terms: Now, let's plug the values of tan⁡(π12)\tan\left(\frac{\pi}{12}\right) and tan⁡(5π3)\tan\left(\frac{5\pi}{3}\right) into the original expression: tan⁡(π12)+tan⁡(5π3)1−tan⁡(π12)tan⁡(5π3)=(3−33+3−3)1−(3−33+3)(−3)\frac{\tan\left(\frac{\pi}{12}\right)+\tan\left(\frac{5\pi}{3}\right)}{1-\tan\left(\frac{\pi}{12}\right)\tan\left(\frac{5\pi}{3}\right)} = \frac{\left(\frac{3 - \sqrt{3}}{3 + \sqrt{3}} - \sqrt{3}\right)}{1 - \left(\frac{3 - \sqrt{3}}{3 + \sqrt{3}}\right)(-\sqrt{3})}.
  8. Final Simplification: Simplify the numerator: (3−3−3)=(3−23)(3 - \sqrt{3} - \sqrt{3}) = (3 - 2\sqrt{3}).
  9. Final Simplification: Simplify the numerator: (3−3−3)=(3−23)(3 - \sqrt{3} - \sqrt{3}) = (3 - 2\sqrt{3}). Simplify the denominator: (1−((3−3)/(3+3))(−3))=(1−(33−3)/(3+3))(1 - ((3 - \sqrt{3}) / (3 + \sqrt{3}))( -\sqrt{3})) = (1 - (3\sqrt{3} - 3) / (3 + \sqrt{3})).
  10. Final Simplification: Simplify the numerator: (3−3−3)=(3−23)(3 - \sqrt{3} - \sqrt{3}) = (3 - 2\sqrt{3}). Simplify the denominator: (1−((3−3)/(3+3))(−3))=(1−(33−3)/(3+3))(1 - ((3 - \sqrt{3}) / (3 + \sqrt{3}))( -\sqrt{3})) = (1 - (3\sqrt{3} - 3) / (3 + \sqrt{3})). Combine the terms in the denominator: (1−(33−3)/(3+3))=((3+3−33+3)/(3+3))(1 - (3\sqrt{3} - 3) / (3 + \sqrt{3})) = ((3 + \sqrt{3} - 3\sqrt{3} + 3) / (3 + \sqrt{3})).
  11. Final Simplification: Simplify the numerator: 3−3−33 - \sqrt{3} - \sqrt{3} = 3−233 - 2\sqrt{3}. Simplify the denominator: 1−((3−3)/(3+3))(−3)1 - ((3 - \sqrt{3}) / (3 + \sqrt{3}))( -\sqrt{3}) = 1−(33−3)/(3+3)1 - (3\sqrt{3} - 3) / (3 + \sqrt{3}). Combine the terms in the denominator: 1−(33−3)/(3+3)1 - (3\sqrt{3} - 3) / (3 + \sqrt{3}) = 3+3−33+33+3\frac{3 + \sqrt{3} - 3\sqrt{3} + 3}{3 + \sqrt{3}}. Simplify the denominator further: 3+3−33+33+3\frac{3 + \sqrt{3} - 3\sqrt{3} + 3}{3 + \sqrt{3}} = 6−233+3\frac{6 - 2\sqrt{3}}{3 + \sqrt{3}}.
  12. Final Simplification: Simplify the numerator: (3−3−3)=(3−23)(3 - \sqrt{3} - \sqrt{3}) = (3 - 2\sqrt{3}). Simplify the denominator: (1−((3−3)/(3+3))(−3))=(1−(33−3)/(3+3))(1 - ((3 - \sqrt{3}) / (3 + \sqrt{3}))( -\sqrt{3})) = (1 - (3\sqrt{3} - 3) / (3 + \sqrt{3})). Combine the terms in the denominator: (1−(33−3)/(3+3))=((3+3−33+3)/(3+3))(1 - (3\sqrt{3} - 3) / (3 + \sqrt{3})) = ((3 + \sqrt{3} - 3\sqrt{3} + 3) / (3 + \sqrt{3})). Simplify the denominator further: ((3+3−33+3)/(3+3))=((6−23)/(3+3))((3 + \sqrt{3} - 3\sqrt{3} + 3) / (3 + \sqrt{3})) = ((6 - 2\sqrt{3}) / (3 + \sqrt{3})). Now we have the simplified expression: ((3−23)/(6−23))((3 - 2\sqrt{3}) / (6 - 2\sqrt{3})).
  13. Final Simplification: Simplify the numerator: (3−3−3)=(3−23)(3 - \sqrt{3} - \sqrt{3}) = (3 - 2\sqrt{3}). Simplify the denominator: (1−((3−3)/(3+3))(−3))=(1−(33−3)/(3+3))(1 - ((3 - \sqrt{3}) / (3 + \sqrt{3}))( -\sqrt{3})) = (1 - (3\sqrt{3} - 3) / (3 + \sqrt{3})). Combine the terms in the denominator: (1−(33−3)/(3+3))=((3+3−33+3)/(3+3))(1 - (3\sqrt{3} - 3) / (3 + \sqrt{3})) = ((3 + \sqrt{3} - 3\sqrt{3} + 3) / (3 + \sqrt{3})). Simplify the denominator further: ((3+3−33+3)/(3+3))=((6−23)/(3+3))((3 + \sqrt{3} - 3\sqrt{3} + 3) / (3 + \sqrt{3})) = ((6 - 2\sqrt{3}) / (3 + \sqrt{3})). Now we have the simplified expression: ((3−23)/(6−23))((3 - 2\sqrt{3}) / (6 - 2\sqrt{3})). We can see that the numerator and the denominator are the same, so the expression simplifies to 11.
  14. Final Simplification: Simplify the numerator: 3−3−33 - \sqrt{3} - \sqrt{3} = 3−233 - 2\sqrt{3}. Simplify the denominator: 1−((3−3)/(3+3))(−3))=(1−(33−3)/(3+3)).Combinethetermsinthedenominator:$1−(33−3)/(3+3)1 - ((3 - \sqrt{3}) / (3 + \sqrt{3}))( -\sqrt{3})) = (1 - (3\sqrt{3} - 3) / (3 + \sqrt{3})). Combine the terms in the denominator: \$1 - (3\sqrt{3} - 3) / (3 + \sqrt{3}) = 3+3−33+33+3\frac{3 + \sqrt{3} - 3\sqrt{3} + 3}{3 + \sqrt{3}}. Simplify the denominator further: 3+3−33+33+3\frac{3 + \sqrt{3} - 3\sqrt{3} + 3}{3 + \sqrt{3}} = 6−233+3\frac{6 - 2\sqrt{3}}{3 + \sqrt{3}}. Now we have the simplified expression: 3−236−23\frac{3 - 2\sqrt{3}}{6 - 2\sqrt{3}}. We can see that the numerator and the denominator are the same, so the expression simplifies to 11. Therefore, the exact value of the original expression is 11, which corresponds to choice (B).

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