Given: #y=cos(sin(x))" [1]"#
Compute #dy/dx#:
#dy/dx = -cos(x)sin(sin(x))#
Compute #(d^2y)/(dx^2)" [2]"#
#(d^2y)/(dx^2) = sin(x) sin(sin(x)) - cos^2(x) cos(sin(x))" [3]"#
Given:
#(d^2y)/(dx^2)+tan(x) dy/dx+y cos^2(x)=0" [4]"#
Substitute equations [1], [2], and [3] into equation [4]:
#sin(x) sin(sin(x)) - cos^2(x) cos(sin(x))+tan(x) (-cos(x)sin(sin(x)))+(cos(sin(x))) cos^2(x)=0#
Please observe that two terms (in red) sum to 0:
#sin(x) sin(sin(x)) color(red)(- cos^2(x) cos(sin(x)))+tan(x) (-cos(x)sin(sin(x)))+color(red)((cos(sin(x))) cos^2(x))=0#
Write the equation without those terms:
#sin(x) sin(sin(x))+tan(x) (-cos(x)sin(sin(x)))=0#
Substitute #tan(x) = sin(x)/cos(x)#:
#sin(x) sin(sin(x))+sin(x)/cos(x) (-cos(x)sin(sin(x)))=0#
Please observe the factors (in red) that cancel by division:
#sin(x) sin(sin(x))+sin(x)/(color(red)(cos(x))) (-color(red)(cos(x))sin(sin(x)))=0#
Write the equation without those factors:
#sin(x) sin(sin(x))-sin(x)sin(sin(x))=0#
The left side sums to 0:
#0=0# Q.E.D.
