2.3.4.2 ✓ Maple. Time used: 0.003 (sec). Leaf size: 17

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

\begin{align*} y = \frac {\arccos \left (-\cos \left (2 x \right )+4 c_1 \right )}{2} \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} \\ {} & {} & \sec \left (x \right )^{2} \tan \left (y \left (x \right )\right ) \left (\frac {d}{d x}y \left (x \right )\right )+\sec \left (y \left (x \right )\right )^{2} \tan \left (x \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 {\sec \left (y \left (x \right )\right )^{2} \tan \left (x \right )}{\sec \left (x \right )^{2} \tan \left (y \left (x \right )\right )} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\left (\frac {d}{d x}y \left (x \right )\right ) \tan \left (y \left (x \right )\right )}{\sec \left (y \left (x \right )\right )^{2}}=-\frac {\tan \left (x \right )}{\sec \left (x \right )^{2}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {\left (\frac {d}{d x}y \left (x \right )\right ) \tan \left (y \left (x \right )\right )}{\sec \left (y \left (x \right )\right )^{2}}d x =\int -\frac {\tan \left (x \right )}{\sec \left (x \right )^{2}}d x +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{2 \sec \left (y \left (x \right )\right )^{2}}=\frac {1}{2 \sec \left (x \right )^{2}}+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \left (x \right ) \\ {} & {} & \left \{y \left (x \right )=2 \pi \_Z1 +\mathrm {arcsec}\left (\frac {1}{\sqrt {-\cos \left (x \right )^{2}-2 \mathit {C1}}}\right ), y \left (x \right )=-\mathrm {arcsec}\left (\frac {1}{\sqrt {-\cos \left (x \right )^{2}-2 \mathit {C1}}}\right )+2 \pi \_Z1 , y \left (x \right )=-\pi +\mathrm {arcsec}\left (\frac {1}{\sqrt {-\cos \left (x \right )^{2}-2 \mathit {C1}}}\right )+2 \pi \_Z1 , y \left (x \right )=2 \pi \_Z1 +\pi -\mathrm {arcsec}\left (\frac {1}{\sqrt {-\cos \left (x \right )^{2}-2 \mathit {C1}}}\right )\right \} \\ \bullet & {} & \textrm {Redefine the integration constant(s)}\hspace {3pt} \\ {} & {} & \left \{y \left (x \right )=2 \pi \_Z1 +\mathrm {arcsec}\left (\frac {1}{\sqrt {-\cos \left (x \right )^{2}+\mathit {C1}}}\right ), y \left (x \right )=-\mathrm {arcsec}\left (\frac {1}{\sqrt {-\cos \left (x \right )^{2}+\mathit {C1}}}\right )+2 \pi \_Z1 , y \left (x \right )=-\pi +\mathrm {arcsec}\left (\frac {1}{\sqrt {-\cos \left (x \right )^{2}+\mathit {C1}}}\right )+2 \pi \_Z1 , y \left (x \right )=2 \pi \_Z1 +\pi -\mathrm {arcsec}\left (\frac {1}{\sqrt {-\cos \left (x \right )^{2}+\mathit {C1}}}\right )\right \} \\ \bullet & {} & \textrm {Assumptions for}\hspace {3pt} \left \{\_Z1 \right \}\hspace {3pt}\textrm {are:}\hspace {3pt} \\ {} & {} & \left \{\_Z1 ::\mathbb {Z}\right \} \end {array} \]