<p>We complete the section method with new simple and versatile techniques to solve some equations that have composite functions as solutions and to study Ulam stability and their hyperstability. We exemplify the malleability of the results obtained by solving equations of the form <Equation ID="Equ32"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1188_Article_Equ32.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="272" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} f\left( \arccos \left| \cos u\cdot \cos v\right| \right) =f\left( u\right) +f\left( v\right) \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>f</mi> <mfenced close=")" open="("> <mo>arccos</mo> <mfenced close="|" open="|"> <mo>cos</mo> <mi>u</mi> <mo>·</mo> <mo>cos</mo> <mi>v</mi> </mfenced> </mfenced> <mo>=</mo> <mi>f</mi> <mfenced close=")" open="("> <mi>u</mi> </mfenced> <mo>+</mo> <mi>f</mi> <mfenced close=")" open="("> <mi>v</mi> </mfenced> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>on relevant real domains, then giving Ulam stability couples and control functions that induce hyperstability for these equations.</p>