Identify an equivalent expression to $\frac{1 - \cos(2x) + \sin(2x)}{1 + \cos(2x) + \sin(2x)}$?
Understand the Problem
The question asks for an equivalent expression to a given trigonometric expression. We need to simplify the given expression using trigonometric identities to match one of the provided options.
Answer
$\cos(x)$
Answer for screen readers
$\cos(x)$
Steps to Solve
- Rewrite $\cot(x)$ in terms of $\sin(x)$ and $\cos(x)$
Recall that $\cot(x) = \frac{\cos(x)}{\sin(x)}$. Substitute this into the given expression: $$ \frac{\cot(x)}{\csc(x)} = \frac{\frac{\cos(x)}{\sin(x)}}{\csc(x)} $$
- Rewrite $\csc(x)$ in terms of $\sin(x)$
Recall that $\csc(x) = \frac{1}{\sin(x)}$. Substitute this into the expression: $$ \frac{\frac{\cos(x)}{\sin(x)}}{\frac{1}{\sin(x)}} $$
- Simplify the complex fraction
To simplify the complex fraction, multiply the numerator by the reciprocal of the denominator: $$ \frac{\cos(x)}{\sin(x)} \cdot \frac{\sin(x)}{1} $$
- Cancel out $\sin(x)$
Since $\sin(x)$ appears in both the numerator and the denominator, we can cancel them out, assuming $\sin(x) \neq 0$: $$ \frac{\cos(x)}{\cancel{\sin(x)}} \cdot \frac{\cancel{\sin(x)}}{1} = \cos(x) $$
Thus, the simplified expression is $\cos(x)$.
$\cos(x)$
More Information
The simplification relies on the fundamental trigonometric identities relating $\cot(x)$, $\csc(x)$, $\sin(x)$, and $\cos(x)$.
Tips
A common mistake is incorrectly recalling or applying the definitions of $\cot(x)$ and $\csc(x)$. Another mistake is making an error when simplifying the complex fraction.
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