In the following match each item given under the column C₁ to its correct answer given under the column C₂:

(a) sin (x + y) sin(x – y)

= (sin x.cos y + sin y.cos x)( sin x.cos y - sin y.cos x)

= sin²x cos²y+ sin y.cos x. sin x.cos y- sin x.cos y sin y.cos x- sin²y cos²x

= sin²x cos²y- sin²y cos²x

= sin²x(1- sin²y)- sin²y(1- sin²x)

= sin²x- sin²x sin²y- sin²y+ sin²x sin²y

= sin²x – sin²y

⇒ (a) matches with (iv)

(b) cos(x + y) cos(x – y)

= (cos x.cos y−sin x.sin y)( cos x.cosy+sin x.siny)

= (cos²x cos²y+ cos x.cos y sin x.siny- sin x.sin y cos x.cosy- sin²x sin²y)

= cos²x cos²y- sin²x sin²y

= cos²x(1- sin²y)-(1- cos²x) sin²y

= cos²x- cos²x sin²y- sin²y+ cos²x sin²y

= cos²x – sin²y

⇒ (b) matches with (i)

(c)

⇒ (c) matches with (ii)

(c)

⇒ (d) matches with (iii)