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Chapter 7: Problem 22

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

Short Answer

Expert verified
The identity is proven as \(\frac{\tan y + \text{cot } y}{\text{csc } y} = \text{sec } y\).

Step by step solution

01

Rewrite Tangent and Cotangent

Rewrite the tangent and cotangent functions in terms of sine and cosine.\(\tan y = \frac{\text{sin } y}{\text{cos } y}\) and \(\text{cot } y = \frac{\text{cos } y}{\text{sin } y}\)
02

Substitute Tangent and Cotangent

Substitute these expressions into the original equation: \[\frac{\frac{\text{sin } y}{\text{cos } y} + \frac{\text{cos } y}{\text{sin } y}}{\text{csc } y}\]Remember that \(\text{csc } y = \frac{1}{\text{sin } y}\)
03

Simplify the Numerators

Combine the terms in the numerator over a common denominator: \[\frac{\frac{\text{sin}^2 y + \text{cos}^2 y}{\text{sin } y \text{cos } y}}{\text{csc } y}\]Since \(\text{sin}^2 y + \text{cos}^2 y = 1\), the expression becomes \[\frac{\frac{1}{\text{sin } y \text{cos } y}}{\text{csc } y}\]
04

Substituting Cosecant

Substitute \(\text{csc } y = \frac{1}{\text{sin } y}\) back into the equation: \[\frac{\frac{1}{\text{sin } y \text{cos } y}}{\frac{1}{\text{sin } y}} = \frac{1}{\text{cos } y}\]This is equal to \(\text{sec } y\).
05

Final Verification

Verify that the simplified expression \(\frac{1}{\text{cos } y}\) is indeed \(\text{sec } y\). The identity is now proven.

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Key Concepts

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

tangent function
Understanding the tangent function is crucial for solving trigonometric identities. The tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side. In terms of sine and cosine functions, tangent can be expressed as: \(\tan y = \frac{\text{sin } y}{\text{cos } y}\). This relationship helps simplify many trigonometric expressions. When breaking down complex identities, substituting \( \tan y \) with \( \frac{\text{sin } y}{\text{cos } y} \) often simplifies the problem.
cotangent function
Cotangent is another important function. It is the reciprocal of the tangent function. This means \( \text{cot } y = \frac{1}{\tan y} \). Alternatively, it can be expressed in terms of sine and cosine as: \( \text{cot } y = \frac{\text{cos } y}{\text{sin } y} \). Knowing this identity helps in trigonometric simplifications. Similar to tangent, substituting cotangent can make otherwise complex-looking equations easier to handle.
trigonometric simplification
Simplifying trigonometric expressions is often about recognizing patterns and relationships between different functions. Here鈥檚 a step-by-step method to simplify complex expressions:
  • Convert Functions: Start by rewriting any tangent and cotangent functions in terms of sine and cosine.
  • Combine Fractions: When you have fractions, try to combine them by finding a common denominator.
  • Use Identities: Use Pythagorean identities such as \( \text{sin}^2 y + \text{cos}^2 y = 1 \).
  • Substitute Back: Substitute back other trigonometric identities (like cosecant, secant).

Remember that practice and familiarity with these steps will improve your ability to simplify and solve trigonometric identities effectively.

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Most popular questions from this chapter

Solve, finding all solutions in \([0,2 \pi)\). $$\cos (\pi-x)+\sin \left(x-\frac{\pi}{2}\right)=1$$

Fill in the blank with the correct term. Some of the given choices will not be used. $$\begin{array}{ll}\text { linear speed } & \text { congruent } \\ \text { angular speed } & \text { circular } \\ \text { angle of elevation } & \text { periodic } \\ \text { angle of depression } & \text { period } \\ \text { complementary } & \text { amplitude } \\ \text { supplementary } & \text { quadrantal } \\ \text { similar } & \text { radian measure }\end{array}$$ __________ is the amount of rotation per unit of time.

Solve using a calculator, finding all solutions in \([0,2 \pi)\). $$\cos x-2=x^{2}-3 x$$

Temperature During an IIIness. The temperature \(T\), in degrees Fahrenheit, of a patient \(t\) days into a 12 -de illness is given by $$T(t)=101.6^{\circ}+3^{\circ} \sin \left(\frac{\pi}{8} t\right)$$ Find the times \(t\) during the illness at which the patient's temperature was \(103^{\circ} .\)

Angles Between Lines. One of the identities gives an easy way to find an angle formed by two lines. Consider two lines with equations \(l_{1}: y=m_{1} x+b_{1}\) and \(l_{2}: y=m_{2} x+b_{2}\) (GRAPH CANNOT COPY) The slopes \(m_{1}\) and \(m_{2}\) are the tangents of the angles \(\theta_{1}\) and \(\theta_{2}\) that the lines form with the positive direction of the \(x\) -axis. Thus we have \(m_{1}=\tan \theta_{1}\) and \(m_{2}=\tan \theta_{2} .\) To find the measure of \(\theta_{2}-\theta_{1},\) or \(\phi,\) we proceed as follows: This formula also holds when the lines are taken in the reverse order. When \(\phi\) is acute, tan \(\phi\) will be positive. When \(\phi\) is obtuse, tan \(\phi\) will be negative. Find the measure of the angle from \(l_{1}\) to \(l_{2}\) $$\begin{aligned} &l_{1}: 2 x=3-2 y\\\ &l_{2}: x+y=5 \end{aligned}$$
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