Rewrite the product as a sum: 10cos (5x) sin (10x)

10cos (5x) sin (10x) = 5[sin (15x) + sin (5x) ]

Step-by-step explanation:

In this question, we are tasked with writing the product as a sum.

To do this, we shall be using the sum to product formula below;

cosαsinβ = 1/2[ sin (α + β) - sin (α - β) ]

From the question, we can say α = 5x and β = 10x

Plugging these values into the equation, we have

10cos (5x) sin (10x) = (10) * 1/2[sin (5x + 10x) - sin (5x - 10x) ]

= 5[sin (15x) - sin (-5x) ]

We apply odd identity i. e sin (-x) = - sinx

Thus applying same to sin (-5x)

sin (-5x) = - sin (5x)

Thus;

5[sin (15x) - sin (-5x) ] = 5[sin (15x) - (-sin (5x)) ]

= 5[sin (15x) + sin (5x) ]

Hence, 10cos (5x) sin (10x) = 5[sin (15x) + sin (5x) ]