What is \(\frac{{\cos {\rm{\theta }}}}{{1 - \tan {\rm{\theta }}}} + \frac{{\sin {\rm{\theta }}}}{{1 - \cot {\rm{\theta }}}}\) equal to?
sin θ + cos θ
Solving the Trigonometric Expression
The question asks us to simplify the given trigonometric expression: \( \frac{{\cos {\rm{\theta }}}}{{1 - \tan {\rm{\theta }}}} + \frac{{\sin {\rm{\theta }}}}{{1 - \cot {\rm{\theta }}}} \).
To simplify this expression, we will convert tangent and cotangent into sine and cosine terms and then combine the fractions.
Recall the fundamental trigonometric identities:
- \( \tan {\rm{\theta }} = \frac{{\sin {\rm{\theta }}}}{{\cos {\rm{\theta }}}} \)
- \( \cot {\rm{\theta }} = \frac{{\cos {\rm{\theta }}}}{{\sin {\rm{\theta }}}} \)
Substitute these into the given expression:
\( \frac{{\cos {\rm{\theta }}}}{{1 - \frac{{\sin {\rm{\theta }}}}{{\cos {\rm{\theta }}}}}} + \frac{{\sin {\rm{\theta }}}}{{1 - \frac{{\cos {\rm{\theta }}}}{{\sin {\rm{\theta }}}}}} \)
Now, simplify the denominators by finding a common denominator in each term:
First term denominator: \( 1 - \frac{{\sin {\rm{\theta }}}}{{\cos {\rm{\theta }}}} = \frac{{\cos {\rm{\theta }}}}{{\cos {\rm{\theta }}}} - \frac{{\sin {\rm{\theta }}}}{{\cos {\rm{\theta }}}} = \frac{{\cos {\rm{\theta }} - \sin {\rm{\theta }}}}{{\cos {\rm{\theta }}}} \)
Second term denominator: \( 1 - \frac{{\cos {\rm{\theta }}}}{{\sin {\rm{\theta }}}} = \frac{{\sin {\rm{\theta }}}}{{\sin {\rm{\theta }}}} - \frac{{\cos {\rm{\theta }}}}{{\sin {\rm{\theta }}}} = \frac{{\sin {\rm{\theta }} - \cos {\rm{\theta }}}}{{\sin {\rm{\theta }}}} \)
Substitute these back into the expression:
\( \frac{{\cos {\rm{\theta }}}}{{\frac{{\cos {\rm{\theta }} - \sin {\rm{\theta }}}}{{\cos {\rm{\theta }}}}}} + \frac{{\sin {\rm{\theta }}}}{{\frac{{\sin {\rm{\theta }} - \cos {\rm{\theta }}}}{{\sin {\rm{\theta }}}}}} \)
When dividing by a fraction, we multiply by its reciprocal:
\( \cos {\rm{\theta }} \cdot \frac{{\cos {\rm{\theta }}}}{{\cos {\rm{\theta }} - \sin {\rm{\theta }}}} + \sin {\rm{\theta }} \cdot \frac{{\sin {\rm{\theta }}}}{{\sin {\rm{\theta }} - \cos {\rm{\theta }}}} \)
\( \frac{{\cos^2 {\rm{\theta }}}}{{\cos {\rm{\theta }} - \sin {\rm{\theta }}}} + \frac{{\sin^2 {\rm{\theta }}}}{{\sin {\rm{\theta }} - \cos {\rm{\theta }}}} \)
Notice that the denominators are almost the same. We can make the second denominator equal to the first by factoring out -1 from \( (\sin {\rm{\theta }} - \cos {\rm{\theta }}) \):
\( (\sin {\rm{\theta }} - \cos {\rm{\theta }}) = -(\cos {\rm{\theta }} - \sin {\rm{\theta }}) \)
Substitute this into the second term:
\( \frac{{\cos^2 {\rm{\theta }}}}{{\cos {\rm{\theta }} - \sin {\rm{\theta }}}} + \frac{{\sin^2 {\rm{\theta }}}}{{-(\cos {\rm{\theta }} - \sin {\rm{\theta }})}} \)
\( \frac{{\cos^2 {\rm{\theta }}}}{{\cos {\rm{\theta }} - \sin {\rm{\theta }}}} - \frac{{\sin^2 {\rm{\theta }}}}{{\cos {\rm{\theta }} - \sin {\rm{\theta }}}} \)
Now that both terms have the same denominator, we can combine the numerators:
\( \frac{{\cos^2 {\rm{\theta }} - \sin^2 {\rm{\theta }}}}{{\cos {\rm{\theta }} - \sin {\rm{\theta }}}} \)
Recognize the numerator as a difference of squares: \( a^2 - b^2 = (a-b)(a+b) \). Here \( a = \cos {\rm{\theta }} \) and \( b = \sin {\rm{\theta }} \).
\( \cos^2 {\rm{\theta }} - \sin^2 {\rm{\theta }} = (\cos {\rm{\theta }} - \sin {\rm{\theta }})(\cos {\rm{\theta }} + \sin {\rm{\theta }}) \)
Substitute this factorization into the expression:
\( \frac{{(\cos {\rm{\theta }} - \sin {\rm{\theta }})(\cos {\rm{\theta }} + \sin {\rm{\theta }})}}{{\cos {\rm{\theta }} - \sin {\rm{\theta }}}} \)
Assuming \( \cos {\rm{\theta }} - \sin {\rm{\theta }} \neq 0 \) (i.e., \( \theta \neq \frac{\pi}{4} + n\pi \), where n is an integer), we can cancel out the common term \( (\cos {\rm{\theta }} - \sin {\rm{\theta }}) \) from the numerator and the denominator:
\( \cos {\rm{\theta }} + \sin {\rm{\theta }} \)
This simplified expression is equal to \( \sin {\rm{\theta }} + \cos {\rm{\theta }} \).
Let's look at the given options:
| Option | Expression |
|---|---|
| 1 | \( \sin {\rm{\theta }} - \cos {\rm{\theta }} \) |
| 2 | \( \sin {\rm{\theta }} + \cos {\rm{\theta }} \) |
| 3 | \( 2\sin {\rm{\theta }} \) |
| 4 | \( 2\cos {\rm{\theta }} \) |
Comparing our simplified expression with the options, we find that it matches option 2.
Revision Table: Key Trigonometric Identities
Understanding fundamental identities is crucial for simplifying trigonometric expressions. Here's a quick review:
| Identity Type | Identity |
|---|---|
| Reciprocal Identity | \( \tan {\rm{\theta }} = \frac{1}{{\cot {\rm{\theta }}}} \) |
| Ratio Identity | \( \tan {\rm{\theta }} = \frac{{\sin {\rm{\theta }}}}{{\cos {\rm{\theta }}}} \) |
| Ratio Identity | \( \cot {\rm{\theta }} = \frac{{\cos {\rm{\theta }}}}{{\sin {\rm{\theta }}}} \) |
| Pythagorean Identity | \( \sin^2 {\rm{\theta }} + \cos^2 {\rm{\theta }} = 1 \) |
| Difference of Squares | \( a^2 - b^2 = (a-b)(a+b) \) |
In this problem, we specifically used the ratio identities for tangent and cotangent and the difference of squares factorization.
Additional Information on Trigonometric Simplification
When simplifying complex trigonometric expressions, a common strategy is to rewrite all terms using only sine and cosine functions. This often makes it easier to combine terms, find common denominators, and apply identities.
Steps typically involved:
- Convert tangent, cotangent, secant, and cosecant into sine and cosine.
- Simplify fractions and combine terms using algebraic techniques (finding common denominators, factoring, etc.).
- Apply fundamental identities (Pythagorean identities, etc.) to further simplify.
- Look for opportunities to factor or expand expressions.
This systematic approach helps in reducing the expression to its simplest form or matching it with one of the given options.
Important Questions from Trigonometric Identities
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