Simplify the trigonometric expression cos(x)/(1 - sin(x)) - tan(x).
Understand the Problem
The question asks us to simplify the given trigonometric expression (\frac{\cos(x)}{1 - \sin(x)} - \tan(x)) and determine to which of the given options it is equivalent. This involves using trigonometric identities to manipulate the expression into a simpler form that matches one of the options.
Answer
$\sec(x)$
Answer for screen readers
$\sec(x)$
Steps to Solve
- Rewrite $\tan(x)$ in terms of $\sin(x)$ and $\cos(x)$
We know that $\tan(x) = \frac{\sin(x)}{\cos(x)}$. Substitute this into the original expression:
$$ \frac{\cos(x)}{1 - \sin(x)} - \frac{\sin(x)}{\cos(x)} $$
- Find a common denominator and combine the fractions
The common denominator is $(1 - \sin(x))\cos(x)$. Combine the fractions:
$$ \frac{\cos^2(x) - \sin(x)(1 - \sin(x))}{(1 - \sin(x))\cos(x)} $$
- Expand and simplify the numerator
Expand the numerator:
$$ \frac{\cos^2(x) - \sin(x) + \sin^2(x)}{(1 - \sin(x))\cos(x)} $$
Recall the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$. Substitute this into the numerator:
$$ \frac{1 - \sin(x)}{(1 - \sin(x))\cos(x)} $$
- Cancel the common factor
Cancel the common factor of $(1 - \sin(x))$ from the numerator and the denominator, provided that $\sin(x) \neq 1$:
$$ \frac{1}{\cos(x)} $$
- Rewrite in terms of $\sec(x)$
Since $\sec(x) = \frac{1}{\cos(x)}$, the simplified expression is:
$$ \sec(x) $$
$\sec(x)$
More Information
The simplified expression is $\sec(x)$, which represents the reciprocal of the cosine function.
Tips
A common mistake is to incorrectly apply trigonometric identities or to make algebraic errors when combining fractions. Another common mistake is failing to recognize the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$.
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