Question

write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. csc x(sin x + cos x)

a. - 2 tan²x
b. 1 + cot x
c. sec x csc x
d. sin x tan x

Explanation:

Step1: Recall the reciprocal identity

Recall that $\csc x=\frac{1}{\sin x}$. So, $\csc x(\sin x+\cos x)=\frac{1}{\sin x}(\sin x + \cos x)$.

Step2: Distribute the fraction

Using the distributive property $\frac{a(b + c)}{d}=\frac{ab}{d}+\frac{ac}{d}$, we have $\frac{1}{\sin x}(\sin x+\cos x)=\frac{\sin x}{\sin x}+\frac{\cos x}{\sin x}$.

Step3: Simplify the fractions

$\frac{\sin x}{\sin x} = 1$ and $\frac{\cos x}{\sin x}=\cot x$. So the expression simplifies to $1+\cot x$.

Answer:

B. $1 + \cot x$