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Chapter 5: Problem 35

Verify each identity. $$\frac{\sec x-\csc x}{\sec x+\csc x}=\frac{\tan x-1}{\tan x+1}$$

Short Answer

Expert verified
Upon substitution of trigonometric identities and simplification of both sides of the equation, we get the same result, showing that the identities are indeed equal.

Step by step solution

01

Apply the Definitions of Trigonometric Functions

Recall that \(\sec(x) = \frac{1}{\cos(x)}\), \(\csc(x) = \frac{1}{\sin(x)}\), and \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). Applying these substitutions, the given identity transforms into \(\frac{1/\cos(x) - 1/\sin(x)}{1/\cos(x) + 1/\sin(x)} = \frac{\sin(x)/\cos(x) - 1}{\sin(x)/\cos(x) + 1}\).
02

Simplify the Expressions

Find common denominator of \(\cos(x)\) and \(\sin(x)\) for both numerator and denominator of the left side of the equation: \(\frac{\sin(x)-\cos(x)}{\sin(x)+\cos(x)} = \frac{\sin(x)/\cos(x) - 1}{\sin(x)/\cos(x) + 1}\). Similarly, for the right side, find common denominator \(\cos(x)\) for both numerator and denominator to get \(\frac{\sin(x)-\cos(x)}{\sin(x)+\cos(x)}\).
03

Show the Equality

Now, after simplification, both sides of the equation are equal. The identity is proven.

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Most popular questions from this chapter

Use a sketch to find the exact value of \(\sec \left(\sin ^{-1} \frac{1}{2}\right)\).

Describe a general strategy for solving each equation. Do not solve the equation. $$2 \sin ^{2} x+5 \sin x+3=0$$

Graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, this indicates the equation is not an identity. In these exercises, find a value of \(x\) for which both sides are defined but not equal. $$\cos \left(x+\frac{\pi}{4}\right)=\cos x+\cos \frac{\pi}{4}$$

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$\cos ^{2} x+2 \cos x-2=0$$

Determine whether each -statement makes sense or does not make sense, and explain your reasoning. I solved \(\cos \left(x-\frac{\pi}{3}\right)=-1\) by first applying the formula for the cosine of the difference of two angles.
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