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. ( 1 + cot x )
b. ( - 2 \tan ^ { 2 } x )
c. ( sin x \tan x )
d. ( sec x csc x )

Explanation:

Step1: Use reciprocal identity

We know 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

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Step3: Simplify the fractions

Since \(\frac{\sin x}{\sin x} = 1\) and \(\frac{\cos x}{\sin x}=\cot x\) (by the quotient identity \(\cot x=\frac{\cos x}{\sin x}\)), we get \(1 + \cot x\).

Answer:

A. \(1+\cot x\)