Problem

16. An equivalent expression to $\frac{\sin x+\cos x}{\csc x+\sec x}$ is

Solution

Step 1 :An equivalent expression to $\frac{\sin x+\cos x}{\csc x+\sec x}$ is

Step 2 :Convert the expression into sine and cosine terms: $\frac{\sin x+\cos x}{\frac{1}{\sin x}+\frac{1}{\cos x}}$

Step 3 :Find a common denominator for the denominator: $\frac{\sin x+\cos x}{\frac{\sin x + \cos x}{\sin x \cos x}}$

Step 4 :Simplify the expression: $\frac{\sin x \cos x(\sin x+\cos x)}{\sin x + \cos x}$

Step 5 :Cancel out the common terms: $\sin x \cos x$

Step 6 :Use the double angle identity: $\frac{\sin(2x)}{2}$

Step 7 :\(\boxed{\frac{\sin(2x)}{2}}\)

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