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