prove that cos^4 12A - sin^4 12A = cos 24A
Answers
Answer:
Given:
The equation \frac{sin 12A}{sin 4A} -\frac{cos 12A}{cos 4A}
sin4A
sin12A
−
cos4A
cos12A
.
To Find:
The value of the given equation.
Calculation:
- Take the LCM of the equation and simplify it:
\frac{sin 12A}{sin 4A} -\frac{cos 12A}{cos 4A} = \frac{sin 12A cos 4A - cos 12A sin 4A}{sin 4A cos 4A}
sin4A
sin12A
−
cos4A
cos12A
=
sin4Acos4A
sin12Acos4A−cos12Asin4A
- Now use the formula sin(x-y) = sinx cosy - cosx siny and multiply both denominator and numerator by 2:
⇒ \frac{sin 12A cos 4A- cos 12A sin 4A}{sin 4A cos 4A} = \frac{2 sin (12A - 4A)}{2 sin 4A cos 4A}
sin4Acos4A
sin12Acos4A−cos12Asin4A
=
2sin4Acos4A
2sin(12A−4A)
- Simplify it and use the formula sin 2x = 2 sinx cosx:
⇒ \frac{2 sin (12A - 4A)}{2 sin 4A cos 4A} = \frac{2 sin 8A}{sin 8A}
2sin4Acos4A
2sin(12A−4A)
=
sin8A
2sin8A
- Cancel out the terms to get the final answer:
⇒ \frac{2 sin 8A}{sin 8A} = 2
sin8A
2sin8A
=2
- So, the value of \frac{sin 12A}{sin 4A} -\frac{cos 12A}{cos 4A} = 2
sin4A
sin12A
−
cos4A
cos12A
=2