2.1.12.5 ✓ Maple. Time used: 0.065 (sec). Leaf size: 10

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

\begin{align*} x = \arcsin \left (\sin \left (1\right ) {\mathrm e}^{t}\right ) \end{align*}

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
<- separable successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d t}x \left (t \right )=\tan \left (x \left (t \right )\right ), x \left (0\right )=1\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 {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d t}x \left (t \right )=\tan \left (x \left (t \right )\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d t}x \left (t \right )}{\tan \left (x \left (t \right )\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {\frac {d}{d t}x \left (t \right )}{\tan \left (x \left (t \right )\right )}d t =\int 1d t +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (\sin \left (x \left (t \right )\right )\right )=t +\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \left (t \right ) \\ {} & {} & \left \{x \left (t \right )=2 \pi \_Z1 +\arcsin \left ({\mathrm e}^{t +\mathit {C1}}\right ), x \left (t \right )=-\arcsin \left ({\mathrm e}^{t +\mathit {C1}}\right )+\pi +2 \pi \_Z1 \right \} \\ \bullet & {} & \textrm {Redefine the integration constant(s)}\hspace {3pt} \\ {} & {} & \left \{x \left (t \right )=2 \pi \_Z1 +\arcsin \left (\mathit {C1} \,{\mathrm e}^{t}\right ), x \left (t \right )=-\arcsin \left (\mathit {C1} \,{\mathrm e}^{t}\right )+\pi +2 \pi \_Z1 \right \} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} x \left (0\right )=1 \\ {} & {} & 1=2 \pi \_Z1 +\arcsin \left (\mathit {C1} \right ) \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} \textit {\_C1} \\ {} & {} & \mathit {C1} =-\sin \left (2 \pi \_Z1 -1\right ) \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \textit {\_C1} =-\sin \left (2 \pi \_Z1 -1\right )\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & x \left (t \right )=2 \pi \_Z1 +\arcsin \left ({\mathrm e}^{t} \sin \left (1\right )\right ) \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} x \left (0\right )=1 \\ {} & {} & 1=-\arcsin \left (\mathit {C1} \right )+\pi +2 \pi \_Z1 \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} \textit {\_C1} \\ {} & {} & \textrm {No solution}\hspace {3pt} \\ \bullet & {} & \textrm {Solution does not satisfy initial condition}\hspace {3pt} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & x \left (t \right )=2 \pi \_Z1 +\arcsin \left ({\mathrm e}^{t} \sin \left (1\right )\right ) \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} \left \{\_Z1 \right \} \\ {} & {} & x \left (t \right )=\arcsin \left ({\mathrm e}^{t} \sin \left (1\right )\right ) \end {array} \]