Question

establish the identity. tan u cot u - sin²u = cos²u
rewrite the factor cot u in the first term, tan u cot u, in terms of tan u. tan u·1/tan u - sin²u
simplify the expression from the previous step by canceling the common factor. - sin²u

Explanation:

Step1: Recall cot - tan relationship

We know that $\cot u=\frac{1}{\tan u}$, so $\tan u\cot u=\tan u\times\frac{1}{\tan u}$.

Step2: Simplify the product

$\tan u\times\frac{1}{\tan u} = 1$. So the left - hand side of the identity becomes $1-\sin^{2}u$.

Step3: Use the Pythagorean identity

By the Pythagorean identity $\sin^{2}u+\cos^{2}u = 1$, we can rewrite $1-\sin^{2}u$ as $\cos^{2}u$, which is the right - hand side of the given identity.

Answer:

$1$