Sol. The equations of two curves are

y = f(x) ... (i)

and, y = f (x) sin x ... (ii)

Solving these two equations, we get

f(x) = f(x). sin x

⇒ sin x = 1 [  f(x) > 0]

⇒ x = 2n π + , n ∈ Z

⇒ x = (4n +1) , n ∈ Z

Differentiating (i) and (ii), w.r.t. x, we get

= f ′ (x) and = f ′ (x) sin x + f(x) cos x

∴ m 1 = Slope of the tangent to (i) at x = (4n + 1)

= f ′

and,

m 2 = Slope of the tangent to (ii) at x = (4n + 1)

= f ′ sin + f

cos (4n +1)

= f ′

Clearly, m 1 = m 2 . Hence, the two curves have common tangents at common points.