Question

# express (tan t + sin t) / (1+sec t) in terms of a single trig function

Trigonometric Functions
express $$\displaystyle\frac{{{\tan{{t}}}+{\sin{{t}}}}}{{{1}+{\sec{{t}}}}}$$ in terms of a single trig function

We are given: $$\displaystyle\frac{{{\tan{{t}}}+{\sin{{t}}}}}{{{1}+{\sec{{t}}}}}$$
Express using sine and cosine: $$=\frac{\left(\frac{\sin t}{\cos t}\right)+\sin t}{1+\left(\frac{1}{\cos t}\right)} =\frac{\frac{(\sin t+\sin t \cos t)}{\cos t}}{\frac{\cos t+1}{ \cos t}} =\frac{\sin t+\sin t \cos t}{\cos t+1}$$
Factor out sint: $$\displaystyle={\sin{{t}}}\frac{{{1}+{\cos{{t}}}}}{{{\cos{{t}}}+{1}}}$$
Cancel $$1+\cos tt: =\sin t$$