What is \(\frac{{\cos \theta }}{{1 + \sin \theta }} + \frac{1}{{\cot \theta }}\) equal to?
sec θ
Simplifying Trigonometric Expressions
Let's simplify the given trigonometric expression step-by-step. The expression we need to simplify is:
\(\frac{{\cos \theta }}{{1 + \sin \theta }} + \frac{1}{{\cot \theta }}\)
First, recall the reciprocal identity for cotangent:
\(\frac{1}{{\cot \theta }} = \tan \theta\)
So, we can rewrite the expression as:
\(\frac{{\cos \theta }}{{1 + \sin \theta }} + \tan \theta\)
Next, express \(\tan \theta\) in terms of sine and cosine using the identity \(\tan \theta = \frac{{\sin \theta }}{{\cos \theta }}\):
\(\frac{{\cos \theta }}{{1 + \sin \theta }} + \frac{{\sin \theta }}{{\cos \theta }}\)
Now, we need to combine these two fractions by finding a common denominator. The common denominator is \((1 + \sin \theta)(\cos \theta)\). We cross-multiply the numerators with the denominators of the opposite fractions:
- Multiply the numerator of the first fraction (\(\cos \theta\)) by the denominator of the second fraction (\(\cos \theta\)).
- Multiply the numerator of the second fraction (\(\sin \theta\)) by the denominator of the first fraction (\(1 + \sin \theta\)).
This gives us the combined fraction:
\(\frac{{\cos \theta \cdot \cos \theta + \sin \theta \cdot (1 + \sin \theta)}}{{(1 + \sin \theta)(\cos \theta)}}\)
Simplify the numerator:
\(\frac{{\cos^2 \theta + \sin \theta + \sin^2 \theta}}{{(1 + \sin \theta)(\cos \theta)}}\)
Recall the fundamental trigonometric identity: \(\sin^2 \theta + \cos^2 \theta = 1\). We can substitute this into the numerator:
\(\frac{{(\cos^2 \theta + \sin^2 \theta) + \sin \theta}}{{(1 + \sin \theta)(\cos \theta)}}\)
\(\frac{{1 + \sin \theta}}{{(1 + \sin \theta)(\cos \theta)}}\)
Assuming that \(1 + \sin \theta \neq 0\), we can cancel the term \((1 + \sin \theta)\) from both the numerator and the denominator:
\(\frac{{\cancel{{1 + \sin \theta}}}}{{\cancel{{(1 + \sin \theta)}} (\cos \theta)}}\)
This simplifies to:
\(\frac{1}{{\cos \theta}}\)
Finally, recall the reciprocal identity for cosine:
\(\frac{1}{{\cos \theta}} = \sec \theta\)
So, the simplified form of the expression \(\frac{{\cos \theta }}{{1 + \sin \theta }} + \frac{1}{{\cot \theta }}\) is \(\sec \theta\).
Let's check the options provided:
- cosec \(\theta\)
- sec \(\theta\)
- sec \(\theta\) + cosec \(\theta\)
- cosec \(\theta\) – cot \(\theta\)
Our simplified expression \(\sec \theta\) matches option 2.
| Step | Expression | Reason |
|---|---|---|
| 1 | \(\frac{{\cos \theta }}{{1 + \sin \theta }} + \frac{1}{{\cot \theta }}\) | Original expression |
| 2 | \(\frac{{\cos \theta }}{{1 + \sin \theta }} + \tan \theta\) | Using \(\frac{1}{{\cot \theta }} = \tan \theta\) |
| 3 | \(\frac{{\cos \theta }}{{1 + \sin \theta }} + \frac{{\sin \theta }}{{\cos \theta }}\) | Using \(\tan \theta = \frac{{\sin \theta }}{{\cos \theta }}\) |
| 4 | \(\frac{{\cos^2 \theta + \sin \theta (1 + \sin \theta)}}{{(1 + \sin \theta)(\cos \theta)}}\) | Combining fractions with common denominator |
| 5 | \(\frac{{\cos^2 \theta + \sin \theta + \sin^2 \theta}}{{(1 + \sin \theta)(\cos \theta)}}\) | Expanding numerator |
| 6 | \(\frac{{1 + \sin \theta}}{{(1 + \sin \theta)(\cos \theta)}}\) | Using \(\cos^2 \theta + \sin^2 \theta = 1\) |
| 7 | \(\frac{1}{{\cos \theta}}\) | Canceling \((1 + \sin \theta)\) (assuming \(1 + \sin \theta \neq 0\)) |
| 8 | \(\sec \theta\) | Using \(\frac{1}{{\cos \theta}} = \sec \theta\) |
Revision Table: Key Trigonometric Identities
| Identity Type | Identity |
|---|---|
| Reciprocal Identity | \(\cot \theta = \frac{1}{{\tan \theta}}\) |
| Reciprocal Identity | \(\sec \theta = \frac{1}{{\cos \theta}}\) |
| Ratio Identity | \(\tan \theta = \frac{{\sin \theta }}{{\cos \theta }}\) |
| Pythagorean Identity | \(\sin^2 \theta + \cos^2 \theta = 1\) |
Additional Information on Trigonometric Simplification
Simplifying trigonometric expressions is a fundamental skill in trigonometry. It often involves using identities to rewrite expressions in a simpler or more standard form. Here are some tips for simplifying trigonometric expressions:
- Convert to Sine and Cosine: Often, converting all terms to sine and cosine makes it easier to find common denominators and apply identities.
- Use Pythagorean Identities: Identities like \(\sin^2 \theta + \cos^2 \theta = 1\), \(1 + \tan^2 \theta = \sec^2 \theta\), and \(1 + \cot^2 \theta = \csc^2 \theta\) are very useful for substitution.
- Factor Expressions: Look for opportunities to factor out common terms or use algebraic identities (like difference of squares).
- Find Common Denominators: When adding or subtracting fractions, combine them over a common denominator.
- Recognize Patterns: Become familiar with common trigonometric identities and look for them within the expression.
- Work on One Side: If simplifying to match another expression or value, work on only one side of the equation until it matches the other.
Practice is key to mastering trigonometric simplification.
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