(1) Given [cos(A - B)] / [cos(A + B)] = -9/7, find the value of tan A tan B
(2) Given [sin(A - B)] / [sin(A + B)] = 2, find the value of tan A cot B
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Answer:
(1)
[cos(A - B)] / [cos(A + B)] = -9 / 7
[cos A cos B + sin A sin B] / [cos A cos B - sin A sin B] = -9/7
1 + [2 sin A sin B] / [cos A cos B - sin A sin B] = -9/7
[2 sin A sin B] / [cos A cos B - sin A sin B] = -9/7 - 1 = -16/7
[sin A sin B] / [cos A cos B - sin A sin B] = -8/7
Thus,
[cos A cos B - sin A sin B] / [sin A sin B] = -7/8
cot A cot B - 1 = -7/8
cot A cot B = 1/8
tan A tan B = 8
(2)
[sin(A - B)] / [sin(A + B)] = 2
[sin A cos B - cos A sin B] / [sin A cos B + cos A sin B] = 2
1 - [2 cos A sin B] / [sin A cos B + cos A sin B] = 2
[2 cos A sin B] / [sin A cos B + cos A sin B] = -1
[cos A sin B] / [sin A cos B + cos A sin B] = -1/2
Flip over
[sin A cos B + cos A sin B] / [cos A sin B] = -2
tan A cot B + 1 = -2
tan A cot B = -3
