Multiple choice

If $\theta$ is an acute angle and $\tan \theta+\cot \theta=2$, then $\tan^7 \theta +\cot^7 \theta=2$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If tan(theta) + cot(theta) = 2, then tan(theta) = 1 (since x + 1/x = 2 implies x=1). If tan(theta) = 1, then cot(theta) = 1. 1^7 + 1^7 = 2. The statement is true.

AI explanation

Using the algebraic identity that cot(theta) = 1 / tan(theta), the given equation becomes tan(theta) + 1 / tan(theta) = 2. Multiplying the entire equation by tan(theta) gives the quadratic equation tan^2(theta) - 2 * tan(theta) + 1 = 0. Factoring this yields (tan(theta) - 1)^2 = 0, meaning tan(theta) = 1. Since cot(theta) is the reciprocal of tan(theta), cot(theta) is also 1, which makes tan^7(theta) + cot^7(theta) equal to 1^7 + 1^7 = 2. Therefore, the statement is true.

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