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Magazine Chapter 8: Problem 58
Establish each identity. $$\frac{\cot \theta}{1-\tan \theta}+\frac{\tan \theta}{1-\cot \theta}=1+\tan \theta+\cot \theta$$
Short Answer
Expert verified
The identity is established.
Step by step solution
01
Express cotangent and tangent in terms of sine and cosine
Recall that \(\tan \theta = \frac{\text{sin} \theta}{\text{cos} \theta}\) and \(\text{cot} \theta = \frac{\text{cos} \theta}{\text{sin} \theta}\). Substitute these expressions into the given equation.
02
Rewrite the left-hand side
Rewrite \(\frac{\text{cot} \theta}{1-\text{tan} \theta} + \frac{\text{tan} \theta}{1-\text{cot} \theta}\) using the substitutions: \(\frac{\frac{\text{cos} \theta}{\text{sin} \theta}}{1 - \frac{\text{sin} \theta}{\text{cos} \theta}} + \frac{\frac{\text{sin} \theta}{\text{cos} \theta}}{1 - \frac{\text{cos} \theta}{\text{sin} \theta}} \). Simplify the denominators by factoring out the common denominator.
03
Simplify the fractions
Simplify each fraction: \(\frac{\frac{\text{cos} \theta}{\text{sin} \theta}}{\frac{\text{cos} \theta - \text{sin} \theta}{\text{cos} \theta}} + \frac{\frac{\text{sin} \theta}{\text{cos} \theta}}{\frac{\text{sin} \theta - \text{cos} \theta}{\text{sin} \theta}} \). This reduces to: \(\frac{\text{cos} \theta}{\text{cos} \theta - \text{sin} \theta} + \frac{\text{sin} \theta}{\text{sin} \theta - \text{cos} \theta} \).
04
Combine the terms
Since \(\text{cos} \theta - \text{sin} \theta = - (\text{sin} \theta - \text{cos} \theta)\), the fractions become: \(\frac{\text{cos} \theta}{\text{cos} \theta - \text{sin} \theta} - \frac{\text{sin} \theta}{\text{cos} \theta - \text{sin} \theta} \).
05
Add the fractions
Combine the numerators: \(\frac{\text{cos} \theta - \text{sin} \theta}{\text{cos} \theta - \text{sin} \theta} = 1 \).
06
Compare both sides
The left-hand side simplifies to 1. Adding \(\text{tan} \theta + \text{cot} \theta = \frac{\text{sin} \theta}{\text{cos} \theta} + \frac{\text{cos} \theta}{\text{sin} \theta} \) matches the right-hand side. Thus, the identity is established.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Cotangent
Cotangent, represented as \(\text{cot} \theta\), is the reciprocal of tangent. It's defined as the ratio of the adjacent side to the opposite side in a right triangle. More precisely, it can be expressed as the quotient of the cosine and sine functions: \(\text{cot} \theta = \frac{\cos \theta}{\sin \theta}\). Using reciprocal identities allows us to simplify complex trigonometric expressions more easily.
For instance, as seen in the given solution, transforming \(\text{cot} \theta\) to \(\frac{\cos \theta}{\sin \theta}\) makes it easier to simplify the equations and see relationships between the trigonometric functions.
For instance, as seen in the given solution, transforming \(\text{cot} \theta\) to \(\frac{\cos \theta}{\sin \theta}\) makes it easier to simplify the equations and see relationships between the trigonometric functions.
Understanding Tangent
Tangent, denoted as \(\text{tan} \theta\), is fundamentally the ratio of the opposite side to the adjacent side in a right triangle. It can be articulated mathematically as: \(\text{tan} \theta = \frac{\sin \theta}{\cos \theta}\). This is handy for simplifying trigonometric expressions and solving identities.
In our exercise, replacing \(\text{tan} \theta\) with \(\frac{\sin \theta}{\cos \theta}\) enables clearer understanding and simplification of the equation. The symmetrical nature of sine and cosine helps in visualizing and verifying trigonometric identities.
In our exercise, replacing \(\text{tan} \theta\) with \(\frac{\sin \theta}{\cos \theta}\) enables clearer understanding and simplification of the equation. The symmetrical nature of sine and cosine helps in visualizing and verifying trigonometric identities.
Simplifying Expressions
Simplifying trigonometric expressions is a crucial skill in solving identities and other mathematical problems. By substituting trigonometric ratios like \(\text{tan} \theta\) and \(\text{cot} \theta\), the complexity is reduced.
In the provided solution, we first rewrite the original expression using the definitions \(\text{cot} \theta = \frac{\cos \theta}{\sin \theta}\) and \(\text{tan} \theta = \frac{\sin \theta}{\cos \theta}\). Simplifying these and then combining like terms helps us to ultimately show that both sides of the identity are equal. By breaking the problem into smaller, manageable steps, we ensure a clear path to the solution.
Here鈥檚 a quick recap:
In the provided solution, we first rewrite the original expression using the definitions \(\text{cot} \theta = \frac{\cos \theta}{\sin \theta}\) and \(\text{tan} \theta = \frac{\sin \theta}{\cos \theta}\). Simplifying these and then combining like terms helps us to ultimately show that both sides of the identity are equal. By breaking the problem into smaller, manageable steps, we ensure a clear path to the solution.
Here鈥檚 a quick recap:
- Transform cotangent and tangent into sine and cosine.
- Simplify the fractions.
- Combine terms carefully and verify.
