f {(cos5A/cosA) + (sin5A/sinA)} = a +b Cos4A, then find the value of a, b & (a + b)
Answers
Given: {(cos5A/cosA) + (sin5A/sinA)} = a +b Cos4A
To find: the value of a, b & (a + b)
Solution:
- Now we have given:
{(cos5A/cosA) + (sin5A/sinA)} = a +b Cos4A
- Solving LHS, we get:
sin A cos 5A + sin 5A cos A / cos A sin A = a +b cos 4A
- Now we know the formula :
sin ( A + B ) = sin A cos B + cos A sin B .......................(I)
- Applying this, we get:
sin (A + 5A) / cos A sin A = a +b cos 4A
sin (6A) / cos A sin A = a +b cos 4A
- Mulyiplying by 2 in numerator and denominator, we get:
2 sin (6A) / 2 cos A sin A = a +b cos 4A
2 sin (2A + 4A) / sin 2A = a +b cos 4A
- Using (I), we get:
2 { sin 2A cos 4A + cos 2A sin 4A } / sin 2A = a +b cos 4A
2 sin 2A cos 4A / sin 2A + 2 cos 2A sin 4A / sin 2A = a +b cos 4A
2 cos 4A + 2 cos 2A (sin 2(2A)) / sin 2A = a +b cos 4A
2 cos 4A + 2 cos 2A (2 sin 2A cos 2A )/ sin 2A = a +b cos 4A
2 cos 4A + 2 cos 2A (2 cos 2A ) = a +b cos 4A
2 cos 4A + 2(2 cos^2 2A) = a +b cos 4A
2 cos 4A + 2(cos4A + 1) = a +b cos 4A
2 cos 4A + 2cos4A + 2 = a +b cos 4A
4 cos 4A + 2 = a +b cos 4A
- Now comparing the terms, we get:
a = 2, b = 4 and a+b = 6
Answer:
So the value of a is 2, b is 4 and a+b is 6.