Question

to prove this please help

Answer 1

Given,

$y=\sin(\sin x)\\\;\\\text{diff. both sides w.r.t x}\\\;\\y_1=\cos(\sin x)\frac{\sin x}{dx}\\\;\\y_1=\cos(\sin x)\cos x\\\;\\y_1\tan x=\tan x.cos\sin x.\cos x\\\;\\y_1\tan x=\sin x.\cos\sin x............(i)\\\;\\\text{Diff. both sides w.r.t. x}\\\;\\y_2=\cos x\frac{d\cos\sin x}{dx}+cos\sin x\frac{dcos x}{dx}\\\;\\y_2=-cosx^{2}.\sin\sin x -cos\sin x.\sin x .....................(ii)\\\;\\L.H.S.\to\\\;\\y_2+y_1\tan x+ycos^{2}x\\\;\\=-cosx^{2}.\sin\sin x -cos\sin x.\sin x+\sin x.\cos\sin x+cos^{2}x.\sin\sin x\\\;\\=0$