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Magazine Chapter 7: Problem 21
Verify the identity. $$ \frac{\sin u-\sin v}{\sin u+\sin v}=\frac{\tan \frac{1}{2}(u-v)}{\tan \frac{1}{2}(u+v)} $$
Short Answer
Expert verified
The identity is verified: both sides are equal.
Step by step solution
01
Express Left Side in Terms of Tan Half-Angle Formula
Using the sine difference identity, express \( \sin u - \sin v \) and \( \sin u + \sin v \) with half-angle identities. By the half-angle identity, \( \sin u - \sin v = 2 \cos\left(\frac{u+v}{2}\right) \sin\left(\frac{u-v}{2}\right) \) and \( \sin u + \sin v = 2 \sin\left(\frac{u+v}{2}\right) \cos\left(\frac{u-v}{2}\right) \).
02
Simplify the Expression
Substitute these identities into the original expression on the left side: \[ \frac{2 \cos\left(\frac{u+v}{2}\right) \sin\left(\frac{u-v}{2}\right)}{2 \sin\left(\frac{u+v}{2}\right) \cos\left(\frac{u-v}{2}\right)} \]The '2's cancel out, leaving:\[ \frac{\cos\left(\frac{u+v}{2}\right)}{\sin\left(\frac{u+v}{2}\right)} \cdot \frac{\sin\left(\frac{u-v}{2}\right)}{\cos\left(\frac{u-v}{2}\right)} \]
03
Convert Fraction to Tangent
Recognize that \( \frac{\cos\left(\frac{u+v}{2}\right)}{\sin\left(\frac{u+v}{2}\right)} = \cot\left(\frac{u+v}{2}\right) \) and \( \frac{\sin\left(\frac{u-v}{2}\right)}{\cos\left(\frac{u-v}{2}\right)} = \tan\left(\frac{u-v}{2}\right) \). Hence, the expression \[ \frac{\tan\left(\frac{u-v}{2}\right)}{\tan\left(\frac{u+v}{2}\right)} \] matches the right-hand side.
04
Conclusion
Both sides of the equation are identical as we have shown both to be equal to \( \frac{\tan\left(\frac{u-v}{2}\right)}{\tan\left(\frac{u+v}{2}\right)} \). Therefore, the given trigonometric identity is verified.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Half-Angle Identities
When working with trigonometric identities, knowing the half-angle identities can be very helpful. These identities relate the trigonometric functions of half an angle to the functions of the angle itself. In the exercise, the half-angle identities are used to break down more complex expressions for simplification. The basic forms of these identities are as follows:
- For sine: \( \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}} \)
- For cosine: \( \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}} \)
- For tangent: \( \tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \)
Sine Difference Identity
Another powerful tool in trigonometry is the sine difference identity. The identity states:
- \( \sin(u - v) = \sin u \cos v - \cos u \sin v \)
Tangent Function
The tangent function is fundamental to trigonometry, often representing the ratio of the sine and cosine of an angle. Formally, it is given by:
- \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
- \( \tan\left(\frac{u - v}{2}\right) = \frac{\sin\left(\frac{u - v}{2}\right)}{\cos\left(\frac{u - v}{2}\right)} \)
Cotangent Function
As the reciprocal of the tangent function, the cotangent function is also a key player in trigonometry. It is defined as:
- \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)
- \( \frac{\cos\left(\frac{u + v}{2}\right)}{\sin\left(\frac{u + v}{2}\right)} = \cot\left(\frac{u + v}{2}\right) \)
