simplify \\frac{1+\\sec(t)}{1+\\cos(t)} to a single trig function.

Explanation:

Step1: Substitute $\sec(t)=\frac{1}{\cos(t)}$

$$\frac{1+\sec(t)}{1 + \cos(t)}=\frac{1+\frac{1}{\cos(t)}}{1+\cos(t)}$$

Step2: Simplify the numerator

$$\frac{1+\frac{1}{\cos(t)}}{1+\cos(t)}=\frac{\frac{\cos(t)+ 1}{\cos(t)}}{1+\cos(t)}$$

Step3: Divide by a fraction

$$\frac{\frac{\cos(t)+1}{\cos(t)}}{1+\cos(t)}=\frac{\cos(t)+1}{\cos(t)}\cdot\frac{1}{1+\cos(t)}$$

Step4: Cancel out the common factor

$$\frac{\cos(t)+1}{\cos(t)}\cdot\frac{1}{1+\cos(t)}=\frac{1}{\cos(t)}$$

Answer:

$\sec(t)$