If $\displaystyle \theta $ is an acute angle and $\displaystyle \tan \theta+\cot \theta =2 $ then
- $\displaystyle \tan \theta =\frac{1}{2} $
- $\displaystyle \tan \theta=2 $
- $\displaystyle \tan^{5} \theta+\cot ^{5}\theta=32 $
- $\displaystyle \tan^{7} \theta+\cot ^{7}\theta=2 $
If tan(theta) + cot(theta) = 2, then tan(theta) + 1/tan(theta) = 2. Let x = tan(theta), so x + 1/x = 2, which implies x^2 - 2x + 1 = 0, or (x - 1)^2 = 0. Thus, tan(theta) = 1, which means theta = 45 degrees. Consequently, tan^n(theta) + cot^n(theta) = 1^n + 1^n = 2 for any n.
Let x equal tan theta, so the given equation x plus 1 over x equals 2 can be multiplied by x to form the quadratic equation x squared minus 2x plus 1 equals 0. Factoring this gives the square of x minus 1 equals 0, meaning x equals 1 and therefore tan theta equals 1. Since cot theta is the reciprocal of tan theta, cot theta also equals 1. Substituting these values into the expression in the last option gives 1 to the power of 7 plus 1 to the power of 7, which equals 2.