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Chapter 7: Problem 4

Prove the identity. $$\frac{1+\tan y}{1+\cot y}=\frac{\sec y}{\csc y}$$

Short Answer

Expert verified
Both sides simplify to \(\frac{\text{sin} y}{\text{cos} y}\). The identity is proven.

Step by step solution

01

Express \(\tan y\) and \(\text{cot} y\) in terms of \(\text{sin} y\) and \(\text{cos} y\)

Recall that \(\tan y = \frac{\text{sin} y}{\text{cos} y} \) and \(\text{cot} y = \frac{\text{cos} y}{\text{sin} y} \). Substitute these expressions into the left-hand side of the given identity: \(\frac{1+\frac{\text{sin} y}{\text{cos} y}}{1+\frac{\text{cos} y}{\text{sin} y}}\).
02

Simplify the numerator and the denominator

Combine terms in the numerator and the denominator: \(\frac{\frac{\text{cos} y + \text{sin} y}{\text{cos} y}}{\frac{\text{sin} y + \text{cos} y}{\text{sin} y}}\).
03

Simplify the complex fraction

Rewrite the complex fraction by multiplying the numerator and the denominator by \(\text{cos} y \times \text{sin} y\): \(\frac{(\text{cos} y + \text{sin} y) \times \text{sin} y}{(\text{sin} y + \text{cos} y) \times \text{cos} y}\). This simplifies to \(\frac{\text{sin} y (\text{cos} y + \text{sin} y)}{\text{cos} y (\text{sin} y + \text{cos} y)}\).
04

Cancel common factors

Observe that \(\text{cos} y + \text{sin} y\) is a common factor in both the numerator and the denominator. Cancel it out to obtain: \(\frac{\text{sin} y}{\text{cos} y}\).
05

Express the simplified fraction using trigonometric identities

Recognize that \(\frac{\text{sin} y}{\text{cos} y} = \tan y\). To match this with the right-hand side of the equation, express \(\frac{\text{sec} y}{\text{csc} y}\) using \(\text{sec} y = \frac{1}{\text{cos} y}\) and \(\text{csc} y = \frac{1}{\text{sin} y}\), giving: \(\frac{1/\text{cos} y}{1/\text{sin} y} = \frac{\text{sin} y}{\text{cos} y}\).
06

Prove the identity

Since both sides simplify to \(\frac{\text{sin} y}{\text{cos} y}\), this proves that the identity \(\frac{1+\tan y}{1+\text{cot} y} = \frac{\text{sec} y}{\text{csc} y}\) is true.

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Key Concepts

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

Tangent (tan)
The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. In a formula, it's written as: \ \(\tan y = \frac{\sin y}{\cos y}\)\
This means that tangent combines both sine and cosine. \Remember: \
    \
  • \(\sin y\) is the ratio of the opposite side to the hypotenuse.
    • \
  • \(\cos y\) is the ratio of the adjacent side to the hypotenuse.
    • \
Combining them into tangent gives a key trigonometric function that is very useful in simplifying and solving trigonometric identities.
Cotangent (cot)
Cotangent is essentially the reciprocal of the tangent function. It is defined as: \ \(\cot y = \frac{\cos y}{\sin y}\)\
Cotangent is the ratio of the adjacent side to the opposite side in a right triangle.\
\Remember: \
    \
  • Reciprocal means flipping the numerator and the denominator.
    • \
  • If \(\tan y\) is known, \(\cot y\) is easily found by taking its reciprocal.
    • \
Cotangent often appears in trigonometric identities and is very handy for simplifying expressions, especially when combined with tangent.
Trigonometric Simplification
Trigonometric simplification is the process of using trigonometric identities to make an expression simpler or more manageable. This often involves: \ \
    \
  • Substituting functions with their identities. For example, \( \tan y = \frac{\sin y}{\cos y}\).
    • \
  • Combining like terms.
    • \
  • Canceling common factors.
    • \
In steps 2 to 4 of our solution, we used simplification to turn a complex fraction into a simpler expression: \ \1. Combine terms in the numerator and denominator \ \2. Factor and cancel common terms \ \3. Use reciprocal identities like \(\sec y\) and \(\csc y\) when needed.\ This method helps verify identities and makes solving trigonometric equations easier.
Secant (sec)
Secant is the reciprocal of the cosine function. It is defined as: \ \(\sec y = \frac{1}{\cos y}\)
It represents how the cosine function is supplemented in calculations.
Remember: \ \
  • \(\cos y\) determines the horizontal position on the unit circle.
    • \
  • If \(\cos y\) is known, \(\sec y\) is found by taking its reciprocal.
    • \
\In the given identity, \(\sec y\) appears in the right-hand side fraction. Recognizing secant helps transform and simplify trigonometric expressions effectively.
Cosecant (csc)
Cosecant is the reciprocal of the sine function. It is written as: \ \ \(\csc y = \frac{1}{\sin y}\) \ \
It is the ratio of the hypotenuse to the opposite side. \
Remember: \ \
    \
  • \(\sin y\) determines the vertical position on the unit circle.
    • \
  • If \(\sin y\) is known, finding \(\csc y\) is as simple as taking its reciprocal.
    • \
In our identity problem, recognizing and using \(\csc y\) helps align both sides of the given identity to the same trigonometric form.

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