Problem 27 - Precalculus Essentials

Question

Verify each identity. $$\frac{\sin x-\sin y}{\sin x+\sin y}=\tan \frac{x-y}{2} \cot \frac{x+y}{2}$$

Step-by-Step Solution

Answer

The identity is verified, as the rearranged left-hand side expression matches the right-hand side.

Step 1: Express sin in terms of cos

Let's begin by using the identity \(\sin \theta = \cos (90 - \theta)\) to transform the expression. LHS will be \(\frac{\cos (90 - x) - \cos(90 - y)}{\cos (90 - x) + \cos (90 - y)}\).

Step 2: Apply the formula for cos difference

Now, we can apply the difference formula for cosine to the LHS, \(\cos A -\cos B = -2 \sin \left(\frac{A + B}{2}\right) \sin \left(\frac{B - A}{2}\right)\), we get \(- \frac{2\sin \left(\frac{x + y}{2}\right) \sin \left(\frac{y - x}{2}\right)}{2\cos \left(\frac{x + y}{2}\right) \cos \left(\frac{y - x}{2}\right)}\). Simplifying, we get \( -\tan \left(\frac{x - y}{2}\right) \tan \left(\frac{x + y}{2}\right)\).

Step 3: Apply the tan negation

This expression is still not identical to the RHS, so let's apply the identity \(\tan(-\theta)= -\tan(\theta)\) to get \( \tan \left(\frac{x - y}{2}\right) \tan \left(\frac{x + y}{2}\right)\).

Step 4: Apply the tan to cot conversion

Finally, use the identity \(\tan (\theta) = \cot(90 - \theta)\) to convert the expression \(\tan \left(\frac{x - y}{2}\right)\) to \(\cot \left(\frac{90 -(x - y)}{2}\right)\) and \(\tan \left(\frac{x + y}{2}\right)\) to \(\cot \left(\frac{90 -(x + y)}{2}\right)\). After this conversion, the converted LHS becomes \(\cot \left(\frac{x - y}{2}\right) \tan \left(\frac{x + y}{2}\right)\), which matches the RHS. Hence, the identity is verified.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry. It helps us understand the relation of an angle in a right triangle to the length of the side opposite that angle. If we consider a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse, which is the longest side of the triangle.

In mathematical terms, this is expressed as:\[ \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \]Here are a few important properties of the sine function:

• It ranges from -1 to 1.
• It is periodic, repeating every 360 degrees or \(2\pi\) radians.
• The sine of 0 degrees is 0, and the sine of 90 degrees is 1.

Aside from its use in triangles, the sine function appears in various mathematical identities, such as the double angle formulas and the angle addition and subtraction identities. In the problem, we can use sine's relation to cosine via the identity \( \sin \theta = \cos (90 - \theta) \), to facilitate transformations and simplify the given trigonometric expression.

Understanding sine can also steer you through complex expressions by enabling conversion between sine and cosine, both necessary for resolving many trigonometric proofs and verifications.

Cosine Function

The cosine function complements the sine function and plays a pivotal role in trigonometry. It represents the ratio of the length of the adjacent side to the hypotenuse in a right triangle:
\[ \cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} \]Much like its sine counterpart, the cosine function ranges between -1 and 1, repeating every 360 degrees or \(2\pi\) radians. Important points to remember include:

• \(\cos 0\) is 1 while \(\cos 90\) is 0.
• Cosine is an even function, meaning \(\cos(-\theta) = \cos \theta\).
• It helps in defining the unit circle, where each point \((x,y)\) on the circle is \((\cos \theta, \sin \theta)\).

The homework problem involves multiple transformations and identities using cosine. One such transformation is using the cosine of complementary angles, allowing transitions between sine and cosine, as shown by \( \sin \theta = \cos(90 - \theta) \). Additionally, the identity for cos differences, \( \cos A - \cos B = -2 \sin \left(\frac{A + B}{2}\right) \sin \left(\frac{B - A}{2}\right) \), is crucial in turning complex trigonometric expressions into manageable equations, critical for verifying trigonometric identities.

Tangent Function

The tangent function is another useful trigonometric function that represents the ratio of the sine function to the cosine function. In a right triangle, tangent of an angle is expressed as:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\text{opposite side}}{\text{adjacent side}} \]The tangent function, however, diverges from sine and cosine as it ranges from negative infinity to positive infinity and repeats every 180 degrees or \( \pi \) radians. Some key features of the tangent function include:

• The tangent of 45 degrees or \( \frac{\pi}{4} \) radians is 1.
• It is undefined where cosine is zero because division by zero is undefined; this occurs at 90 degrees and its odd multiples.
• Being an odd function, \( \tan(-\theta) = -\tan(\theta) \).

In the problem, tangent helps convert complex expressions due to its behavior and corresponding identities. For example, the negation identity \( \tan(-\theta) = -\tan(\theta) \) is useful for simplifying expressions, as seen in step 3 of the solution. Moreover, its relation to cotangent is employed, as \( \tan( \theta ) = \cot(90 - \theta) \), for converting and verifying identities. The tangent function's flexibility in intersections between sine and cosine components is indispensable for solution processes involving trigonometric identities.