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Chapter 5: Problem 19

Verify the identity. $$\frac{\cot ^{2} t}{\csc t}=\frac{1-\sin ^{2} t}{\sin t}$$

Short Answer

Expert verified
The identity \(\frac{\cot ^{2} t}{\csc t}=\frac{1-\sin ^{2} t}{\sin t}\) has been verified by rewriting 'cot' and 'csc' in terms of 'sin' and 'cos' and applying the Pythagorean identity.

Step by step solution

01

Rewrite cot and csc in terms of sin and cos

To start, it's easier to work with the simpler trigonometric functions 'sin' and 'cos'. So, replace 'cot' with \(\frac{cos t}{sin t}\) and 'csc' with \(\frac{1}{sin t}\) in on the left-hand side (LHS) of the equation. This gives us: \(\frac{{\left( \frac{cos t}{sin t} \right)} ^{2}}{\frac{1}{sin t}}\).
02

Simplify the expression

We can simplify the expression by multiplying the numerator and the denominator of the fraction by \(sin^2 t\). This removes the denominators inside the brackets and changes the expression to \(\frac{cos^2 t}{sin t}\). We should use the opportunity to simplify the fraction with their equivalent forms in terms of 'sin' as it makes the next steps easier to solve.
03

Apply Pythagorean identity on cos^2 t

We can use the Pythagorean identity \(\cos^2 t + \sin^2 t = 1\) to express \(\cos^2 t\) as \((1 - \sin^2 t)\). Substituting in the equation, we get \(\frac{1 - \sin^2 t}{\sin t}\) which is exactly the right-hand side (RHS) of the equation. This verifies the identity.

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Key Concepts

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

Cotangent
The cotangent, often abbreviated as "cot," is a trigonometric function that serves as the reciprocal of the tangent function. In mathematical terms, it is written as \( \cot t = \frac{\cos t}{\sin t} \). Here, \( \cos t \) represents the cosine of angle \( t \) and \( \sin t \) represents the sine of angle \( t \).
To better understand cotangent, consider its basic properties:
  • \( \cot t \) is undefined when \( \sin t = 0 \), which happens at multiples of \( \pi \).
  • \( \cot t \) is periodic with a period of \( \pi \).
  • In a unit circle, cotangent is the ratio of the adjacent side to the opposite side.
Understanding cotangent is crucial in trigonometry, especially when simplifying expressions or solving trigonometric identities, as seen in the exercise where cotangent was expressed in terms of sine and cosine to simplify the equation.
Cosecant
The cosecant function, abbreviated as "csc," is another reciprocal trigonometric function. It is defined as the reciprocal of the sine function: \( \csc t = \frac{1}{\sin t} \). This means that it depends directly on the sine of an angle.
Here are some key characteristics of the cosecant:
  • \( \csc t \) is undefined where \( \sin t = 0 \), similar to \( \cot t \), which is why it is rarely used in everyday trigonometric calculations.
  • Like sine, \( \csc t \) has a period of \( 2\pi \).
  • Cosecant is typically greater than or equal to 1 or less than or equal to -1, never falling between -1 and 1.
In the given problem, the cosecant function is replaced with its reciprocal identity involving sine to simplify the left-hand side of the equation. By expressing it as \( \frac{1}{\sin t} \), we are able to make calculations easier, ultimately leading towards proving the identity.
Pythagorean Identity
The Pythagorean identity is a fundamental relation in trigonometry, defined as \( \cos^2 t + \sin^2 t = 1 \). It plays a crucial role in simplifying expressions and verifying identities, like the one in the exercise.
Key aspects of the Pythagorean identity include:
  • You can rearrange it to express \( \cos^2 t \) or \( \sin^2 t \) in terms of each other, such as \( \cos^2 t = 1 - \sin^2 t \) or \( \sin^2 t = 1 - \cos^2 t \).
  • This identity is derived from the Pythagorean theorem applied in a unit circle context.
In the problem, the Pythagorean identity was applied to transform \( \cos^2 t \) into \( 1 - \sin^2 t \). This step was pivotal to obtaining an expression that matches the right-hand side of the identity we were trying to verify. By using this identity, we managed to simplify the trigonometric expression significantly, proving the identity to be correct.

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