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Chapter 7: Problem 77

Verify that each trigonometric equation is an identity. $$\sec x-\cos x+\csc x-\sin x-\sin x \tan x=\cos x \cot x$$

Short Answer

Expert verified
The equation \( \sec x - \cos x + \csc x - \sin x - \sin x \, \tan x = \cos x \, \cot x \) is an identity.

Step by step solution

01

Express all trigonometric functions in terms of sine and cosine

\( \sec x = \frac{1}{\cos x} \), \( \csc x = \frac{1}{\sin x} \), \( \tan x = \frac{\sin x}{\cos x} \), and \( \cot x = \frac{\cos x}{\sin x} \).
02

Rewrite the left-hand side using the identities

Substitute the trigonometric functions:\( \sec x - \cos x + \csc x - \sin x - \sin x \tan x = \frac{1}{\cos x} - \cos x + \frac{1}{\sin x} - \sin x - \sin x \frac{\sin x}{\cos x} \)
03

Simplify the left-hand side

Combine like terms:\( \frac{1}{\cos x} - \cos x + \frac{1}{\sin x} - \sin x - \frac{\sin^2 x}{\cos x} \).
04

Combine fractions

Combine terms with common denominators:\( \frac{1 - \cos^2 x}{\cos x} + \frac{1 - \sin^2 x}{\sin x} \).
05

Simplify using Pythagorean identities

Use \( 1 - \cos^2 x = \sin^2 x \) and \( 1 - \sin^2 x = \cos^2 x \):\( \frac{\sin^2 x}{\cos x} + \frac{\cos^2 x}{\sin x} \).
06

Separate the fractions and simplify

\( \sin x \frac{\sin x}{\cos x} + \cos x \frac{\cos x}{\sin x} \) gives \( \sin x \tan x + \cos x \cot x \). Since \( \sin x \tan x = \sin x \frac{\sin x}{\cos x} = \frac{\sin^2 x}{\cos x} \) and \( \cos x \cot x = \cos x \frac{\cos x}{\sin x} = \frac{\cos^2 x}{\sin x} \), it simplifies to \( \frac{\sin^2 x}{\cos x} + \frac{\cos^2 x}{\sin x} \).
07

Equality of the simplified expressions

Since \( \cos x \cot x \) on the right-hand side is \( \cos x \frac{\cos x}{\sin x} = \frac{\cos^2 x}{\sin x} \), both sides are equal.

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Key Concepts

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

Trigonometric Functions
Understanding trigonometric functions is essential in verifying identities like the one in the exercise. Trigonometric functions relate angles to the ratios of sides of a right triangle. The primary functions to remember are:

  • \( \sin x \) (sine),
  • \( \cos x \) (cosine),
  • \( \tan x \) (tangent) which can be written as \( \frac{\sin x}{\cos x} \).

Other important functions include:

  • \( \csc x \) (cosecant) = \(\frac{1}{\sin x} \),
  • \( \sec x \) (secant) = \(\frac{1}{\cos x} \),
  • \( \cot x \) (cotangent) = \( \frac{\cos x}{\sin x} \).

Understanding these conversions is key in simplifying and verifying trigonometric identities.
Pythagorean Identities
Next, we move on to the Pythagorean identities. These are derived from the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is the sum of the squares of the other two sides. The main Pythagorean identity is:

\[ \sin^2 x + \cos^2 x = 1 \]
From this, we can derive:

  • \( 1 - \sin^2 x = \cos^2 x \),
  • \( 1 - \cos^2 x = \sin^2 x \).

These identities help simplify expressions, particularly in the given exercise. They allow us to replace complex parts of the equation with their simpler equivalents.
Simplifying Expressions
Finally, simplifying expressions is crucial in solving trigonometric equations. To simplify, we often:
  • Express all functions in terms of sine and cosine.
  • Combine like terms.
  • Use common denominators to combine fractions.
  • Apply Pythagorean identities to simplify further.

For example, in the exercise:

\( \sec x - \cos x + \csc x - \sin x - \sin x \tan x = \frac{1}{\cos x} - \cos x + \frac{1}{\sin x} - \sin x - \sin x \frac{\sin x}{\cos x} \)

becomes

\( \frac{1 - \cos^2 x}{\cos x} + \frac{1 - \sin^2 x}{\sin x} \)

Applying Pythagorean identities, we get:

\( \frac{\sin^2 x}{\cos x} + \frac{\cos^2 x}{\sin x} \), which further simplifies to \( \sin x \tan x + \cos x \cot x \).

Recognizing and applying these steps ensures the accuracy of solving trigonometric identities.

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Most popular questions from this chapter

Use identities to write each expression as a single function of \(x\) or \(\theta\). $$\tan \left(180^{\circ}+\theta\right)$$

(This discussion applies to functions of both angles and real numbers.) Consider the following. $$\begin{aligned}\cos (&\left.180^{\circ}-\theta\right) \\\&=\cos 180^{\circ} \cos \theta+\sin 180^{\circ} \sin \theta \\\&=(-1) \cos \theta+(0) \sin \theta \\\&=-\cos \theta\end{aligned}$$ \(\cos \left(180^{\circ}-\theta\right)=-\cos \theta\) is an example of a reduction formula, which is an identity that reduces a function of a quadrantal angle plus or minus \(\theta\) to a function of \(\theta\) alone. Another example of a reduction formula is \(\cos \left(270^{\circ}+\theta\right)=\sin \theta\) Here is an interesting method for quickly determining a reduction formula for a trigonometric function \(f\) of the form \(f(Q \pm \theta),\) where \(Q\) is a quadrantal angle. There are two cases to consider, and in each case, think of \(\boldsymbol{\theta}\) as a small positive angle in order to determine the quadrant in which \(Q \pm \theta\) will lie. Case 1 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along the \(x\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a small positive angle \(\theta .\) If the given function \(f\) is positive in that quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that quadrant, use a - sign. The reduced form will have that \(\operatorname{sign}, f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE GRAPH Case 2 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along the \(y\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a small positive angle \(\theta .\) If the given function \(f\) is positive in that quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that quadrant, use a - sign. The reduced form will have that sign, the cofunction of \(f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE GRAPH Use these ideas to write reduction formulas for each of the following. $$\sin \left(180^{\circ}+\theta\right)$$

Verify that each equation is an identity. $$(\cos 2 x+\sin 2 x)^{2}=1+\sin 4 x$$

Answer each question. Suppose you are solving a trigonometric equation for solutions over the interval \([0,2 \pi)\) and your work leads to \(\frac{1}{2} x=\frac{\pi}{16}, \frac{5 \pi}{12}, \frac{5 \pi}{8} .\) What are the corresponding values of \(x ?\)

(This discussion applies to functions of both angles and real numbers.) Consider the following. $$\begin{aligned}\cos (&\left.180^{\circ}-\theta\right) \\\&=\cos 180^{\circ} \cos \theta+\sin 180^{\circ} \sin \theta \\\&=(-1) \cos \theta+(0) \sin \theta \\\&=-\cos \theta\end{aligned}$$ \(\cos \left(180^{\circ}-\theta\right)=-\cos \theta\) is an example of a reduction formula, which is an identity that reduces a function of a quadrantal angle plus or minus \(\theta\) to a function of \(\theta\) alone. Another example of a reduction formula is \(\cos \left(270^{\circ}+\theta\right)=\sin \theta\) Here is an interesting method for quickly determining a reduction formula for a trigonometric function \(f\) of the form \(f(Q \pm \theta),\) where \(Q\) is a quadrantal angle. There are two cases to consider, and in each case, think of \(\boldsymbol{\theta}\) as a small positive angle in order to determine the quadrant in which \(Q \pm \theta\) will lie. Case 1 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along the \(x\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a small positive angle \(\theta .\) If the given function \(f\) is positive in that quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that quadrant, use a - sign. The reduced form will have that \(\operatorname{sign}, f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE GRAPH Case 2 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along the \(y\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a small positive angle \(\theta .\) If the given function \(f\) is positive in that quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that quadrant, use a - sign. The reduced form will have that sign, the cofunction of \(f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE GRAPH Use these ideas to write reduction formulas for each of the following. $$\tan \left(270^{\circ}-\theta\right)$$
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