Rewrite this expression so that the numerator and denominator each contain a single fraction.
[tex]=\frac{\frac{\cos ^2 x-\sin ^2 x}{\sin ^2 x}}{\frac{\cos ^2 x+\sin ^2 x}{\sin ^2 x}} \text { (Do not simplify) }[/tex]
Re-enter the numerator, and simplify the denominator of this complex fraction.
[tex]=\frac{\square}{\square}[/tex]
Answer (2)
The numerator remains as s i n 2 x c o s 2 x − s i n 2 x .
Simplify the denominator using the trigonometric identity cos 2 x + sin 2 x = 1 , resulting in s i n 2 x 1 .
The rewritten expression is s i n 2 x 1 s i n 2 x c o s 2 x − s i n 2 x .
Therefore, the final answer is s i n 2 x 1 s i n 2 x c o s 2 x − s i n 2 x .
Explanation
Understanding the Problem We are given a complex fraction and asked to rewrite it so that the numerator and denominator each contain a single fraction. We need to re-enter the numerator as is and simplify the denominator using the trigonometric identity cos 2 x + sin 2 x = 1 .
Identifying the Numerator The given expression is: s i n 2 x c o s 2 x + s i n 2 x s i n 2 x c o s 2 x − s i n 2 x The numerator is s i n 2 x c o s 2 x − s i n 2 x , which we will re-enter without simplification.
Simplifying the Denominator The denominator is s i n 2 x c o s 2 x + s i n 2 x . We know the trigonometric identity cos 2 x + sin 2 x = 1 . Therefore, the denominator simplifies to s i n 2 x 1 .
Rewriting the Expression So, the rewritten expression is: s i n 2 x 1 s i n 2 x c o s 2 x − s i n 2 x Thus, the numerator is s i n 2 x c o s 2 x − s i n 2 x and the denominator is s i n 2 x 1 .
Examples
Complex fractions are used in various fields like physics and engineering to simplify complex equations. For example, when calculating electrical impedance in AC circuits, you often encounter complex fractions that need simplification to find the equivalent impedance. Similarly, in fluid dynamics, complex fractions can arise when dealing with flow rates and pressure drops in interconnected pipes. Simplifying these fractions makes the calculations more manageable and provides a clearer understanding of the underlying relationships.
The numerator remains as s i n 2 x c o s 2 x − s i n 2 x , while the denominator simplifies to s i n 2 x 1 . Therefore, the complex fraction is rewritten as s i n 2 x 1 s i n 2 x c o s 2 x − s i n 2 x .
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