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Chapter 5: Problem 48

Verify each identity. $$\frac{\csc t-1}{\cot t}=\frac{\cot t}{\csc t+1}$$

Short Answer

Expert verified
By following these steps, it can be observed that \( \frac{\csc t-1}{\cot t} = \frac{\cot t}{\csc t+1} \) is indeed a valid identity as starting from the LHS and going through the steps we end up with the RHS

Step by step solution

01

Rewrite csc and cot as reciprocal identities

The first step is to rewrite all trigonometric functions in terms of sine and cosine. This will make it easier to simplify the expression. So, rewrite the expression as follows: \[ \frac{1/\sin t - 1}{\cos t / \sin t} = \frac{\cos t / \sin t}{1/\sin t + 1} \]
02

Simplify each side separately

Now, simplifying each side, we get \[ \frac{\sin t - 1}{\cos t} = \frac{\cos t }{\sin t + 1} \]
03

Multiplication by conjugate

Multiplying each side of the equation by the conjugate of the denominator to get rid of the fractions. This means multiplying the LHS of the equation by (\(\sin t + 1\)) and the RHS by \((\cos t + 1)\) we get: \[ \frac{\sin t - 1}{\cos t} \cdot (\sin t + 1) = \frac{\cos t}{\sin t + 1} \cdot (\cos t + 1) \] Upon simplifying we get, \[ \sin^2 t - 1 = \cos^2 t - 1\]
04

Use Pythagorean identity

By using the Pythagorean identity, \(\sin^2 t + \cos^2 t = 1\), we can replace \(\sin^2 t\) with \(1 - \cos^2 t\) and \(\cos^2 t\) with \(1 - \sin^2 t\). This leads us to: \[1 - \cos^2 t - 1 = 1 -\sin^2 t - 1\] which simplifies to \( \cos^2 t = \sin^2 t \)

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Most popular questions from this chapter

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. $$\sin x=-\cos x \tan (-x)$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning.After using an identity to determine the exact value of \(\sin 105^{\circ}, 1\) verified the result with a calculator.

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.

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. $$5 \sec ^{2} x-10=0$$

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. $$\sin (x+\pi)=\sin x$$
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