If the angle $\alpha$ lies in the first quadrant and $\tan \alpha + \cot \alpha =2,$ then the value of $\sqrt{\tan \alpha}+\sqrt{\cot \alpha}$ is

Rewrite the given equation tan alpha plus cot alpha equals 2 by expressing it in terms of sine and cosine, yielding (sin squared alpha plus cos squared alpha) divided by (sin alpha times cos alpha) equals 2. Using the Pythagorean identity, the numerator becomes 1, so sin alpha times cos alpha equals 1 half. The expression to evaluate is the square of (square root of tan alpha plus square root of cot alpha), which expands to tan alpha plus cot alpha plus 2 times the square root of (tan alpha times cot alpha). Substituting the given sum of 2 and the identity cot alpha times tan alpha equals 1 results in 2 plus 2, so the square root of this 4 is 2.

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