2.2.1.2 ✓ Maple. Time used: 0.003 (sec). Leaf size: 11

ode:=sin(x)*cos(y(x))^2+cos(x)^2*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y = -\arctan \left (\sec \left (x \right )+c_1 \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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \sin \left (x \right ) \cos \left (y \left (x \right )\right )^{2}+\cos \left (x \right )^{2} \left (\frac {d}{d x}y \left (x \right )\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-\frac {\sin \left (x \right ) \cos \left (y \left (x \right )\right )^{2}}{\cos \left (x \right )^{2}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d x}y \left (x \right )}{\cos \left (y \left (x \right )\right )^{2}}=-\frac {\sin \left (x \right )}{\cos \left (x \right )^{2}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {\frac {d}{d x}y \left (x \right )}{\cos \left (y \left (x \right )\right )^{2}}d x =\int -\frac {\sin \left (x \right )}{\cos \left (x \right )^{2}}d x +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \tan \left (y \left (x \right )\right )=-\frac {1}{\cos \left (x \right )}+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \left (x \right ) \\ {} & {} & y \left (x \right )=\arctan \left (\frac {\mathit {C1} \cos \left (x \right )-1}{\cos \left (x \right )}\right )+\pi \_Z1 \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y \left (x \right )=\arctan \left (\mathit {C1} -\sec \left (x \right )\right )+\pi \_Z1 \\ \bullet & {} & \textrm {Assumptions for}\hspace {3pt} \left \{\_Z1 \right \}\hspace {3pt}\textrm {are:}\hspace {3pt} \\ {} & {} & \left \{\_Z1 ::\mathbb {Z}\right \} \end {array} \]