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Consider the following expression: (tan((5pi)/(4))+tan((7pi)/(12)))/(1-tan((5pi)/(4))tan((7pi)/(12))) Find the exact value of the expression without a calculator. [Hint: This diagram of special trigonometry. values may help.] Choose 1 answer: (A) (sqrt3)/(3) (B) sqrt3 (C) -(sqrt3)/(3) (D) -sqrt3

Consider the following expression:\newlinetan⁡(5π4)+tan⁡(7π12)1−tan⁡(5π4)tan⁡(7π12) \frac{\tan \left(\frac{5 \pi}{4}\right)+\tan \left(\frac{7 \pi}{12}\right)}{1-\tan \left(\frac{5 \pi}{4}\right) \tan \left(\frac{7 \pi}{12}\right)} \newlineFind the exact value of the expression without a calculator.\newline[Hint: This diagram of special trigonometry. values may help.]\newlineChoose 11 answer:\newline(A) 33 \frac{\sqrt{3}}{3} \newline(B) 3 \sqrt{3} \newline(C) −33 -\frac{\sqrt{3}}{3} \newline(D) −3 -\sqrt{3}

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Full solution

Q. Consider the following expression:\newlinetan⁡(5π4)+tan⁡(7π12)1−tan⁡(5π4)tan⁡(7π12) \frac{\tan \left(\frac{5 \pi}{4}\right)+\tan \left(\frac{7 \pi}{12}\right)}{1-\tan \left(\frac{5 \pi}{4}\right) \tan \left(\frac{7 \pi}{12}\right)} \newlineFind the exact value of the expression without a calculator.\newline[Hint: This diagram of special trigonometry. values may help.]\newlineChoose 11 answer:\newline(A) 33 \frac{\sqrt{3}}{3} \newline(B) 3 \sqrt{3} \newline(C) −33 -\frac{\sqrt{3}}{3} \newline(D) −3 -\sqrt{3}
  1. Use Tangent Addition Formula: Use the tangent addition formula: 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)}.
  2. Calculate Tangent Values: Calculate tan⁡(5π4)\tan\left(\frac{5\pi}{4}\right) and tan⁡(7π12)\tan\left(\frac{7\pi}{12}\right) using special trigonometry values.\newlinetan⁡(5π4)=tan⁡(π+π4)=tan⁡(π4)=−1\tan\left(\frac{5\pi}{4}\right) = \tan\left(\pi + \frac{\pi}{4}\right) = \tan\left(\frac{\pi}{4}\right) = -1 (since tan⁡(π+x)=tan⁡(x)\tan(\pi + x) = \tan(x)).\newlinetan⁡(7π12)=tan⁡(π3+π4)=tan⁡(π3)+tan⁡(π4)1−tan⁡(π3)tan⁡(π4)=3+11−3⋅1\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)} = \frac{\sqrt{3} + 1}{1 - \sqrt{3}\cdot1}.
  3. Plug Values into Expression: Plug these values into the original expression.\newline(tan⁡(5π4)+tan⁡(7π12))/(1−tan⁡(5π4)tan⁡(7π12))=(−1+(3+1))/(1−(−1)(3+1)).(\tan(\frac{5\pi}{4}) + \tan(\frac{7\pi}{12})) / (1 - \tan(\frac{5\pi}{4})\tan(\frac{7\pi}{12})) = (-1 + (\sqrt{3} + 1)) / (1 - (-1)(\sqrt{3} + 1)).
  4. Simplify Numerator and Denominator: Simplify the numerator and denominator.\newline(−1+3+1)/(1−(−3−1))=(3)/(1+3+1)(-1 + \sqrt{3} + 1) / (1 - (-\sqrt{3} - 1)) = (\sqrt{3}) / (1 + \sqrt{3} + 1).
  5. Further Simplify Expression: Further simplify the expression. 32+3\frac{\sqrt{3}}{2 + \sqrt{3}}.
  6. Rationalize Denominator: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator. 3×(2−3)(2+3)×(2−3)\frac{\sqrt{3} \times (2 - \sqrt{3})}{(2 + \sqrt{3}) \times (2 - \sqrt{3})}.
  7. Perform Multiplication: Perform the multiplication. 3×2−3×34−(3)2\frac{\sqrt{3} \times 2 - \sqrt{3} \times \sqrt{3}}{4 - (\sqrt{3})^2}.
  8. Simplify Expression: Simplify the expression. (23−3)/(4−3)(2\sqrt{3} - 3) / (4 - 3).
  9. Finish Simplification: Finish the simplification.\newline(23−3)/1=23−3(2\sqrt{3} - 3) / 1 = 2\sqrt{3} - 3.

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