Multiple choice

If $\displaystyle \sin A=\frac{\sqrt{3}}{2}$ and A is an acute angle, then find the value of $\displaystyle \frac{\tan A-\cot A}{\sqrt{3}+co\sec A}$

  1. $\displaystyle \frac{-2}{5}$
  2. $\displaystyle \frac{2}{5}$
  3. $\displaystyle \frac{2}{3+2\sqrt{3}}$
  4. $-2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If sin A = sqrt(3)/2, A = 60 degrees. tan 60 = sqrt(3), cot 60 = 1/sqrt(3), cosec 60 = 2/sqrt(3). The expression is (sqrt(3) - 1/sqrt(3)) / (sqrt(3) + 2/sqrt(3)). Numerator: (3-1)/sqrt(3) = 2/sqrt(3). Denominator: (3+2)/sqrt(3) = 5/sqrt(3). Result: (2/sqrt(3)) / (5/sqrt(3)) = 2/5.

AI explanation

Since sin A equals root 3 over 2 and A is acute, A is 60 degrees, so tan A is root 3, cot A is 1 over root 3, and cosec A is 2 over root 3. Substituting these into the expression gives (root 3 minus 1 over root 3) divided by (root 3 plus 2 over root 3). Simplifying the numerator gives 2 over root 3, and simplifying the denominator gives 5 over root 3. Dividing the two results yields 2 over 5.

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