Question

If y = cos (sin x), show that: `("d"^2"y")/("dx"^2) + "tan x" "dy"/"dx" + "y"  "cos"^2"x" = 0`

Answer 1

y = cos (sin x)      ....(i)

`"dy"/"dx" = -"sin" ("sin x") "cos x"`       ...(ii)

`("d"^2"y")/("dx"^2)` = sin (sin x) sin x - cos x (cos (sin x)) cos x

= sin (sin x) sin x - cos²x.y       .....(using (i))

= sin (sin x) cos x `((-"sin x")/"cos x")` - y cos²x 

=`"dy"/"dx"` (- tan x) - y cos²x         ......(using(ii))

`("d"^2"y")/("dx"^2) + "tan x" "dy"/"dx" + "y"  "cos"^2"x" = 0`