Question

Simplify the expression: sin (2x) cos (5x) − sin (5x) cos (2x).

Answer 1

Given:

The given expression is $sin (2x) \ cos (5x) - sin (5x) \ cos (2x)$

We need to determine the simplified value of the given expression.

Simplification:

Since, the given expression is in the form of $sin a \ cos b-\cos a \ sin b$, the given expression can be simplified using the identity $\sin (a-b)=\sin a \cos b-\cos a \sin b$

Comparing the given expression with the identity, we get;

$a=2x$ and $b=5x$

Using this in the identity, we get;

$sin (2x) \ cos (5x) - sin (5x) \ cos (2x)=sin(2x-5x)$

Simplifying, we get;

$sin (2x) \ cos (5x) - sin (5x) \ cos (2x)=sin(-3x)$

Thus, the simplified value of the given expression is $sin (-3x)$

Hence, Option c is the correct answer.