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

express $\frac{\mathrm{tan}t+\mathrm{sin}t}{1+\mathrm{sec}t}$ in terms of a single trig function
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Usamah Prosser

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