Question

establish the identity
\\(\frac{4\cot u - \tan u}{\cot u+\tan u}+1 = 5\cos^{2}u\\)

write the left side in terms of sine and cosine
\\(\frac{}{}+1 (do not simplify.)

Explanation:

Step1: Recall cot and tan in terms of sin and cos

Recall that $\cot u=\frac{\cos u}{\sin u}$ and $\tan u = \frac{\sin u}{\cos u}$. Substitute these into the left - hand side fraction:
$$\frac{4\frac{\cos u}{\sin u}-\frac{\sin u}{\cos u}}{\frac{\cos u}{\sin u}+\frac{\sin u}{\cos u}}+1$$

Step2: Get a common denominator for the numerator and denominator of the fraction

The common denominator for the numerator $4\frac{\cos u}{\sin u}-\frac{\sin u}{\cos u}$ is $\sin u\cos u$, so it becomes $\frac{4\cos^{2}u - \sin^{2}u}{\sin u\cos u}$. The common denominator for the denominator $\frac{\cos u}{\sin u}+\frac{\sin u}{\cos u}$ is $\sin u\cos u$, so it becomes $\frac{\cos^{2}u+\sin^{2}u}{\sin u\cos u}$. Then the fraction is $\frac{\frac{4\cos^{2}u - \sin^{2}u}{\sin u\cos u}}{\frac{\cos^{2}u+\sin^{2}u}{\sin u\cos u}}+1$.
Since dividing by a fraction is the same as multiplying by its reciprocal, we have $\frac{4\cos^{2}u - \sin^{2}u}{\cos^{2}u+\sin^{2}u}+1$.

Step3: Use the Pythagorean identity $\sin^{2}u+\cos^{2}u = 1$

Substitute $\sin^{2}u=1 - \cos^{2}u$ into the expression $\frac{4\cos^{2}u-(1 - \cos^{2}u)}{1}+1$.
Expand the numerator: $\frac{4\cos^{2}u - 1+\cos^{2}u}{1}+1=\frac{5\cos^{2}u-1}{1}+1$.

Step4: Simplify the expression

$5\cos^{2}u-1 + 1=5\cos^{2}u$.

For the first part of writing the left - hand side in terms of sine and cosine (the non - simplified form):

Answer:

$$\frac{4\frac{\cos u}{\sin u}-\frac{\sin u}{\cos u}}{\frac{\cos u}{\sin u}+\frac{\sin u}{\cos u}}+1$$