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Chapter 7: Problem 97

Verify that each equation is an identity. $$\frac{2}{1+\cos x}-\tan ^{2} \frac{x}{2}=1$$

Short Answer

Expert verified
The given equation is an identity.

Step by step solution

01

- Use a Trigonometric Identity

Start by using the trigonometric identity for \(\cos 2\theta\): \(\cos x = 1 - 2\sin^2 \frac{x}{2}\). Here, \(2\theta = x\), so substitute into \(\frac{2}{1 + \cos x}\)
02

- Substitute the Identity

Substitute \(\cos x = 1 - 2\sin^2 \frac{x}{2}\) into \(\frac{2}{1 + \cos x}\): \[\frac{2}{1 + (1 - 2\sin^2 \frac{x}{2})} = \frac{2}{2 - 2\sin^2 \frac{x}{2}} = \frac{2}{2(1 - \sin^2 \frac{x}{2})} = \frac{1}{1 - \sin^2 \frac{x}{2}}\]
03

- Simplify Using Algebra

Simplify \(\frac{1}{1 - \sin^2 \frac{x}{2}}\): \[\frac{1}{1 - \sin^2 \frac{x}{2}} = \frac{1}{\cos^2 \frac{x}{2}}= \sec^2 \frac{x}{2}\]
04

- Simplify \tan^2 \frac{x}{2}\

Recall that \sec^2 \theta= 1 + \tan^2 \theta\. Therefore, the equation can be written as: \[ \sec^2 \frac{x}{2} - \tan^2 \frac{x}{2} = 1\]
05

- Verify the Identity

Combine the simplified form into the original expression: \[ \frac{2}{1 + \cos x } = \sec^2 \frac{x}{2}, \sec^2 \frac{x}{2} - \tan^2 \frac{x}{2}= 1\] This confirms the original identity.

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Key Concepts

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

Cosine Double-Angle Identity
The cosine double-angle identity is a key concept in trigonometry. It expresses cosine of a double angle in terms of sine or cosine of a single angle. The formula is:
\[ \cos(2\theta) = 1 - 2 \sin^2(\theta) \]
In the given exercise, this identity is used to simplify expressions involving \cos(x)\. Specifically, by substituting \cos(x) = 1 - 2 \sin^2 \left( \frac{x}{2} \right)\ when x is twice an angle (2胃).
This helps to convert the double-angle cosine term into a format that's easier to manage and simplify.
Secant
The secant function is another crucial concept. It relates to the cosine function and can simplify many trigonometric identities. Secant is defined as:
\[ \sec(\theta) = \frac{1}{\cos(\theta)} \]
In the solution, secant was used through the simplification:
  • Starting with the term \frac{1}{1 - \sin^2 \left( \frac{x}{2} \right)}\,
  • Recognize that \1 - \sin^2 \left( \frac{x}{2} \right) = \cos^2 \left( \frac{x}{2} \right)\,
  • So the term simplifies to \sec^2 \left( \frac{x}{2} \right)\.
This shows the power of recognizing and using secant in trigonometric identities to simplify expressions.
Tangent
Tangent is a fundamental trigonometric function related to sine and cosine. It's defined as:
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
The secant and tangent are related through the identity:
\[ \sec^2(\theta) = 1 + \tan^2(\theta) \]
This relationship was crucial in verifying the given identity. Here's how it was applied:
  • When the expression simplified to \sec^2 \left( \frac{x}{2} \right) \,
  • Recognize from the identity that \sec^2 \left( \frac{x}{2} \right) - \tan^2 \left( \frac{x}{2} \right) = 1 \.
This confirmed that the original equation is, indeed, an identity.

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Most popular questions from this chapter

Explain why attempting to find \(\sin ^{-1} 1.003\) on your calculator will result in an error message.

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. $$-\sec ^{2}(-\theta)+\sin ^{2}(-\theta)+\cos ^{2}(-\theta)$$

Amperage, Wattage, and Voltage Amperage is a measure of the amount of electricity that is moving through a circuit, whereas voltage is a measure of the force pushing the electricity. The wattage \(W\) consumed by an electrical device can be determined by calculating the product of the amperage \(I\) and voltage \(V .\) (Source: Wilcox, G. and C. Hesselberth, Electricity for Engineering Technology, Allyn \& Bacon.)(a) A household circuit has voltage $$ V=163 \sin 120 \pi t $$ when an incandescent light bulb is turned on with amperage $$ I=1.23 \sin 120 \pi t $$ Graph the wattage \(W=V I\) consumed by the light bulb in the window \([0,0.05]\) by $$ [-50,300] $$ (b) Determine the maximum and minimum wattages used by the light bulb. (c) Use identities to determine values for \(a, c,\) and \(\omega\) so that \(W=a \cos (\omega t)+c\) (d) Check your answer by graphing both expressions for \(W\) on the same coordinate axes. (e) Use the graph to estimate the average wattage used by the light. For how many watts (to the nearest integer) do you think this incandescent light bulb is rated?

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\sin \frac{\theta}{2}=1$$

Verify that each trigonometric equation is an identity. $$\frac{1-\cos x}{1+\cos x}=\csc ^{2} x-2 \csc x \cot x+\cot ^{2} x$$
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