Answer:
-4sin(2x)
Step-by-step explanation:
Angle sum identities can be used to simplify this expression.
- cos(a+b) = cos(a)cos(b) -sin(a)sin(b)
- sin(a+b) = sin(a)cos(b) +sin(b)cos(a)
- sin(2x) = 2sin(x)cos(x)
Numerator
The cosine function is even, so cos(-x) = cos(x). The cosine of the sum is ...
cos(90°+x) = cos(90°)cos(x) -sin(90°)sin(x) = -sin(x)
Then the numerator simplifies to ...
4cos(-x)cos(90°+x) = -4cos(x)sin(x) = -2sin(2x)
Denominator
Matching the denominator expression to the sine of a sum relation, we see
sin(30° -x)cos(x) +sin(x)cos(30° -x) = sin((30°-x) +x) = sin(30°) = 1/2
Simplified Expression
The simplified expression is the ratio of the simplified numerator to the simplified denominator:
= -sin(2x)/(1/2)
= -4sin(2x)
