Question

how x=cos theeta
see the pic

Answer 1

in the line before arrow you did a mistake
X sin ( sin ^2 + cos ^2) = sin cos
X sin = sin cos
X = cos

Answer 2

The full question :

x sin θ = y cos θ

Given that :

x sin³θ + y cos³θ = sin θ cos θ

Your steps were wrong .

Here is the correct one :

x sin³ θ + y cos³ θ = sin θ cos θ

⇒ x sin θ ( sin² θ ) + y cos³θ = sin θ cos θ

⇒ y cos θ ( sin²θ ) + y cos³θ = sin θ cos θ

Cancel cos θ both sides :

⇒ y sin²θ + y cos²θ = sin θ

⇒ y ( sin²θ + cos²θ ) = sin θ

Since sin²θ + cos²θ = 1

⇒ y = sin θ

Hence proved.