44.What is the value of [(tan5 A+ tan3 A]/ 4cos4 A (tan5 A – tan3 A)]?
Answers
Given: [(tan5 A+ tan3 A]/ 4cos4 A (tan5 A – tan3 A)]
To find: The value of the given expression.
Solution:
- As we have given the expression in terms of tan A, so lets vonvert it in the form of sin A and cos A.
- After converting, we get:
[(sin 5A/ cos 5A+ sin 3A/ cos 3A]/ (sin 5A/ cos 5A – sin 3A/ cos 3A ) 4cos 4A]
- Now cross multiplying, we get:
{ sin 5A cos 3A + sin 3A cos 5A / cos 5A cos 3A } / { ( sin 5A cos 3A - sin 3A cos 5A / cos 5A cos 3A ) x 4cos 4A }
- So cancelling cos 5A cos 3A , we get:
sin 5A cos 3A + sin 3A cos 5A / (sin 5A cos 3A - sin 3A cos 5A ) x 4cos 4A
- Now we know the identity:
sin(A+B) = sinAcosB + cosAsinB
sin(A-B) = sinAcosB - cosAsinB
sin 2A = 2sinAcosA
- Applying this in the expression, we get:
sin(5A+3A) / sin(5A-3A)x 4cos 4A
sin(8A) / sin(2A) x 4cos 4A
sin 2(4A) / sin(2A) x 4cos 4A
2sin 4A cos 4A / sin(2A) x 4cos 4A
- Cancelling cos 4A, we get:
2sin 4A/sin(2A) x 4
2 sin 2(2A) / sin(2A) x 4
2 x 2sin2A cos2A / sin 2A x 4
- Cancelling sin 2A, we get:
4 cos 2A / 4
cos 2A.
Answer:
So the value of the expression [(tan5 A+ tan3 A]/ 4cos4 A (tan5 A – tan3 A)] is cos 2A.