Given differential equation is, tan‌y‌d‌x+sec2y⋅tan‌x‌d‌y=0 tan‌y‌d‌x=−sec2y‌tan‌x‌d‌y ‌
1
tan‌x
dx=‌
−sec2y
tan‌y
dy ⇒‌‌cot‌x‌d‌x=−‌
sec2y
tan‌y
dy Integrating on both sides, ∫cot‌x‌d‌x=−∫‌
sec2y
tan‌y
dy ⇒‌‌log‌sin‌x=−log‌tan‌y+log‌c ⇒‌‌log‌sin‌x+log‌tan‌y=log‌c ⇒‌log‌sin‌x⋅tan‌y‌‌=log‌c ∴‌sin‌x⋅tan‌y‌‌=c