Multiple choice

If $\theta$ is an acute angle and $\tan {\theta}+\cot {\theta}=2$, find the value of $\tan ^{ 9 }{ \theta } +\cot ^{ 9 }{ \theta } $

  1. $1$
  2. $\sqrt{2}$
  3. $2$
  4. $3$
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C Correct answer
Explanation

tan(theta) + cot(theta) = 2. Since tan(theta) + 1/tan(theta) = 2, tan(theta) must be 1. Thus, theta = 45 degrees. tan^9(45) + cot^9(45) = 1^9 + 1^9 = 1 + 1 = 2.

AI explanation

Multiplying the equation tangent theta plus cotangent theta = 2 by tangent theta gives tangent squared theta plus 1 = 2 tangent theta. This forms the quadratic equation tangent squared theta minus 2 tangent theta plus 1 = 0, which factors to (tangent theta minus 1) squared = 0. This means tangent theta equals 1, so cotangent theta is also 1. Therefore, the value of tangent to the power of 9 theta plus cotangent to the power of 9 theta is 1 to the power of 9 plus 1 to the power of 9, which equals 2.

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