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Chapter 7: Problem 108

Verify that each equation is an identity. $$\frac{\sin (x+y)}{\cos (x-y)}=\frac{\cot x+\cot y}{1+\cot x \cot y}$$

Short Answer

Expert verified
The identity \( \frac{\sin (x+y)}{\cos (x-y)} = \frac{\cot x + \cot y}{1 + \cot x \cot y} \) is verified.

Step by step solution

01

Understand the given equation

The equation to verify is \( \frac{\sin (x+y)}{\cos (x-y)} = \frac{\cot x + \cot y}{1 + \cot x \cot y} \). We need to show that both sides of the equation are equivalent.
02

Express cotangent in terms of sine and cosine

Recall that \( \cot x = \frac{\cos x}{\sin x} \) and \( \cot y = \frac{\cos y}{\sin y} \). Substitute these into the right-hand side of the given equation.
03

Simplify the right-hand side

Simplify \( \frac{\cot x + \cot y}{1 + \cot x \cot y} \) by substituting the expressions from the previous step: \[ \frac{\frac{\cos x}{\sin x} + \frac{\cos y}{\sin y}}{1 + \left( \frac{\cos x}{\sin x} \right) \left( \frac{\cos y}{\sin y} \right)} = \frac{\frac{\cos x \sin y + \cos y \sin x}{\sin x \sin y}}{1 + \frac{\cos x \cos y}{\sin x \sin y}} \]
04

Combine the fractions

Combine the fractions in the numerator and the denominator: \[ \frac{\frac{\cos x \sin y + \cos y \sin x}{\sin x \sin y}}{\frac{\sin x \sin y + \cos x \cos y}{\sin x \sin y}} = \frac{\cos x \sin y + \cos y \sin x}{\sin x \sin y + \cos x \cos y} \]
05

Use trigonometric identities

Recognize that the numerator \( \cos x \sin y + \cos y \sin x \) is the sine of a sum: \( \sin(x+y) \). Also, the denominator \( \sin x \sin y + \cos x \cos y \) is the cosine of a difference: \( \cos(x-y) \). Substitute these back in: \[ \frac{\sin(x+y)}{\cos(x-y)} \]
06

Verify the identity

Now observe that the simplified right-hand side matches the left-hand side. Therefore, it has been verified that \( \frac{\sin (x+y)}{\cos (x-y)} = \frac{\cot x + \cot y}{1 + \cot x \cot y} \) is an identity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cotangent
Cotangent is a trigonometric function related to angles. It is the reciprocal of the tangent function.
The formula for cotangent in terms of sine and cosine is: \(\text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)} \).
This means if we know the sine and cosine of an angle, we can easily find its cotangent.
In our problem, we use cotangent to rewrite terms in a way that simplifies calculations. This is handy when verifying trigonometric identities.
Sine and Cosine Relationships
Sine and cosine are fundamental trigonometric functions that describe relationships in right-angled triangles.
They are defined as follows:
\(\text{sin}(x) = \frac{\text{opposite}}{\text{hypotenuse}} \) and \(\text{cos}(x) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
These functions have important properties and identities that help in transforming and simplifying expressions.
For example, the sine of a sum identity is: \(\text{sin}(x+y) = \text{sin}(x)\text{cos}(y) + \text{cos}(x)\text{sin}(y) \). Likewise, the cosine of a difference identity is: \(\text{cos}(x-y) = \text{cos}(x)\text{cos}(y) + \text{sin}(x)\text{sin}(y) \).
These identities are crucial for manipulating and simplifying expressions, as seen in our step-by-step solution.
Verifying Identities
Verifying trigonometric identities involves proving that two different-looking expressions are, in fact, equal.
Here's a general approach to verify an identity:
  • Start by clearly understanding the given identity.
  • Express all trigonometric functions in terms of sine and cosine, if possible.
  • Simplify both sides of the equation separately, using trigonometric identities and algebraic manipulations.
  • Compare the simplified forms of both sides. If they are equal, the identity is verified.
In our example, we used these steps to show that \(\frac{\text{sin}(x+y)}{\text{cos}(x-y)} = \frac{\text{cot}(x) + \text{cot}(y)}{1 + \text{cot}(x) \text{cot}(y)} \). This involves recognizing and using specific identities like the sine and cosine of sums and differences. In the end, showing that both sides of the equation are equivalent confirms the identity.

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Most popular questions from this chapter

Use identities to write each expression as a single function of \(x\) or \(\theta\). $$\tan (2 \pi-x)$$

The model $$ 0.342 D \cos \theta+h \cos ^{2} \theta=\frac{16 D^{2}}{V_{0}^{2}} $$ is used to reconstruct accidents in which a vehicle vaults into the air after hitting an obstruction. \(V_{0}\) is velocity in feet per second of the vehicle when it hits the obstruction, \(D\) is distance (in feet) from the obstruction to the landing point, and \(h\) is the difference in height (in feet) between landing point and takeoff point. Angle \(\theta\) is the takeoff angle, the angle between the horizontal and the path of the vehicle. Find \(\theta\) to the nearest degree if \(V_{0}=60, D=80,\) and \(h=2\)

Verify that each equation is an identity. $$\tan 2 \theta=\frac{-2 \tan \theta}{\sec ^{2} \theta-2}$$

Write each expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression and all functions are of \(\theta\) only. $$-\sec ^{2}(-\theta)+\sin ^{2}(-\theta)+\cos ^{2}(-\theta)$$

Verify that each trigonometric equation is an identity. $$(1+\sin x+\cos x)^{2}=2(1+\sin x)(1+\cos x)$$
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