2.1.6.4 ✓ Maple. Time used: 0.034 (sec). Leaf size: 13

ode:=diff(x(t),t) = cos(t)/sin(t); 
ic:=[x(1) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\begin{align*} x = \ln \left (\sin \left (t \right )\right )-\ln \left (\sin \left (1\right )\right ) \end{align*}

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
<- quadrature successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d t}x \left (t \right )=\frac {\cos \left (t \right )}{\sin \left (t \right )}, x \left (1\right )=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d t}x \left (t \right ) \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \left (\frac {d}{d t}x \left (t \right )\right )d t =\int \frac {\cos \left (t \right )}{\sin \left (t \right )}d t +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & x \left (t \right )=\ln \left (\sin \left (t \right )\right )+\mathit {C1} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} x \left (1\right )=0 \\ {} & {} & 0=\ln \left (\sin \left (1\right )\right )+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} \textit {\_C1} \\ {} & {} & \mathit {C1} =-\ln \left (\sin \left (1\right )\right ) \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \textit {\_C1} =-\ln \left (\sin \left (1\right )\right )\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & x \left (t \right )=\ln \left (\sin \left (t \right )\right )-\ln \left (\sin \left (1\right )\right ) \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & x \left (t \right )=\ln \left (\sin \left (t \right )\right )-\ln \left (\sin \left (1\right )\right ) \end {array} \]