simplify the expression. sec x sin x + cos(π/2 - x) / 1 + sec x =

Explanation:

Step1: Rewrite secant in terms of cosine

Recall that $\sec x=\frac{1}{\cos x}$. The given expression becomes $\frac{\frac{1}{\cos x}\sin x+\cos(\frac{\pi}{2}-x)}{1 + \frac{1}{\cos x}}$.

Step2: Use the co - function identity

We know that $\cos(\frac{\pi}{2}-x)=\sin x$. So the numerator is $\frac{\sin x}{\cos x}+\sin x=\tan x+\sin x$. The denominator is $\frac{\cos x + 1}{\cos x}$.

Step3: Simplify the complex - fraction

The original expression is $\frac{\frac{\sin x}{\cos x}+\sin x}{\frac{\cos x + 1}{\cos x}}=\frac{\frac{\sin x+\sin x\cos x}{\cos x}}{\frac{\cos x + 1}{\cos x}}$.

Step4: Divide by a fraction

Dividing by a fraction is the same as multiplying by its reciprocal. So we have $\frac{\sin x+\sin x\cos x}{\cos x}\times\frac{\cos x}{\cos x + 1}=\frac{\sin x(1 + \cos x)}{\cos x}\times\frac{\cos x}{\cos x + 1}$.

Step5: Cancel out common factors

Cancel out the common factors $(1 + \cos x)$ and $\cos x$. We get $\sin x$.

Answer:

$\sin x$