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Magazine Chapter 8: Problem 76
Verify the identity. $$ \frac{1}{1-\sin x}-\frac{1}{1+\sin x}=2 \sec x \tan x $$
Short Answer
Expert verified
Yes, \(\frac{1}{1-\sin x}-\frac{1}{1+\sin x}=2 \sec x \tan x\) is an identity.
Step by step solution
01
Find a Common Denominator
To subtract the fractions on the left side, we need a common denominator. The least common denominator for the terms \( \frac{1}{1-\sin x} \) and \( \frac{1}{1+\sin x} \) is \((1-\sin x)(1+\sin x)\).
02
Rewrite Each Fraction
Express each fraction with the common denominator:\[\frac{1}{1-\sin x} = \frac{1+\sin x}{(1-\sin x)(1+\sin x)}\]\[\frac{1}{1+\sin x} = \frac{1-\sin x}{(1-\sin x)(1+\sin x)}\]
03
Subtract the Fractions
Now subtract the two fractions:\[\frac{1+\sin x}{(1-\sin x)(1+\sin x)} - \frac{1-\sin x}{(1-\sin x)(1+\sin x)} = \frac{(1+\sin x) - (1-\sin x)}{(1-\sin x)(1+\sin x)} = \frac{2\sin x}{(1-\sin x)(1+\sin x)}\]
04
Simplify the Denominator Using Identity
The denominator \((1-\sin x)(1+\sin x)\) simplifies to \(1-\sin^2 x\) which is equal to \(\cos^2 x\), using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1\). So, we have:\[\frac{2\sin x}{(1-\sin x)(1+\sin x)} = \frac{2\sin x}{\cos^2 x}\]
05
Use Trigonometric Identities
Split the fraction:\[\frac{2\sin x}{\cos^2 x} = \frac{2\sin x}{\cos x} \cdot \frac{1}{\cos x} = 2\tan x \cdot \sec x\]
06
Conclusion: Verify the Identity
The expression \( \frac{2\sin x}{\cos^2 x} \) simplifies to \(2 \sec x \tan x\), verifying the original identity.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Common Denominators
When dealing with fraction subtraction, finding a common denominator is essential. A common denominator allows us to combine fractions effectively by ensuring they share the same base. In our exercise, we work with two fractions: \( \frac{1}{1-\sin x} \) and \( \frac{1}{1+\sin x} \). To subtract these, a shared denominator is needed. The least common denominator (LCD) here is the product \((1-\sin x)(1+\sin x)\). This step is crucial because it ensures that fractions are comparable, thus making subtraction possible. Here鈥檚 a quick pointer on why common denominators matter:
- Without a common denominator, subtracting fractions directly could yield an incorrect result.
- Finding the least common denominator ensures the equation remains balanced.
- It simplifies further calculations, making the math cleaner.
The Power of the Pythagorean Identity
The Pythagorean identity is one of the cornerstone identities in trigonometry. It states: \( \sin^2 x + \cos^2 x = 1 \).This identity is particularly useful for simplifying expressions involving trigonometric functions. In our problem, we encounter a denominator \((1-\sin x)(1+\sin x)\), which simplifies to \(1-\sin^2 x\). Using the Pythagorean identity, this term simplifies further into \(\cos^2 x\):
- \(1-\sin^2 x = \cos^2 x\) transforms a complex expression into something more manageable.
- This transformation is vital for progressing towards the solution.
- Rewriting expressions via this identity is a common and powerful simplification technique in trigonometry.
Mastering Fraction Subtraction
Subtracting fractions involves a systematic approach once a common denominator is established. In our given exercise, both fractions are expressed with the same denominator \((1-\sin x)(1+\sin x)\). The subtraction then proceeds as:\[\frac{1+\sin x}{(1-\sin x)(1+\sin x)} - \frac{1-\sin x}{(1-\sin x)(1+\sin x)} = \frac{(1+\sin x) - (1-\sin x)}{(1-\sin x)(1+\sin x)}\]This step simplifies to \( \frac{2\sin x}{(1-\sin x)(1+\sin x)} \). Key points to consider:
- Recognition of common terms: Proper setup eliminates redundant processes.
- Adjusting numerators: Subtracting the numerators is straightforward once the common denominator is found.
- Simplicity and accuracy: These go hand-in-hand in fraction subtraction with trigonometric expressions.
Effortless Trigonometric Simplification
The final stretch in this exercise involves trigonometric simplification. After obtaining \( \frac{2\sin x}{\cos^2 x} \), applying identities is the key.We split the fraction and use basic identities:\[\frac{2\sin x}{\cos^2 x} = \frac{2\sin x}{\cos x} \cdot \frac{1}{\cos x} = 2\tan x \cdot \sec x\]Worth noting:
- The use of \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \) identities are instrumental.
- This conversion transforms the fraction into a recognizable form and confirms the identity, fulfilling the exercise's motivation.
- Simplification reduces complexity, making the result readily comparable to the stated identity.
