Rewrite the [tex]$\cot ^2 x$[/tex] terms on the right side of the equation to be in terms of sine and cosine.
[tex]$\cos 2 x=\frac{\frac{\cos ^2 x}{\sin ^2 x}-1}{\frac{\cos ^2 x}{\sin ^2 x}+1}$[/tex]
Apply a Quotient Identity.
Rewrite this expression so that the numerator and denominator each contain a single fraction.
[tex]$=\frac{\square}{\square}$[/tex] (Do not simplify.)
Answer (2)
Rewrite the numerator s i n 2 x c o s 2 x − 1 with a common denominator: s i n 2 x c o s 2 x − s i n 2 x .
Rewrite the denominator s i n 2 x c o s 2 x + 1 with a common denominator: s i n 2 x c o s 2 x + s i n 2 x .
Combine the rewritten numerator and denominator into a single expression.
The final 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 .
Explanation
Understanding the Expression We are given the expression cos 2 x = s i n 2 x c o s 2 x + 1 s i n 2 x c o s 2 x − 1 and we want to rewrite the right-hand side such that both the numerator and the denominator are single fractions.
Rewriting the Numerator First, let's focus on the numerator, which is s i n 2 x c o s 2 x − 1 . To combine these terms into a single fraction, we need a common denominator, which is sin 2 x . Thus, we rewrite 1 as s i n 2 x s i n 2 x . So the numerator becomes s i n 2 x c o s 2 x − s i n 2 x s i n 2 x = s i n 2 x c o s 2 x − s i n 2 x .
Rewriting the Denominator Next, let's focus on the denominator, which is s i n 2 x c o s 2 x + 1 . Similarly, we rewrite 1 as s i n 2 x s i n 2 x . So the denominator becomes s i n 2 x c o s 2 x + s i n 2 x s i n 2 x = s i n 2 x c o s 2 x + s i n 2 x .
Combining the Fractions Now, we can rewrite the entire expression as a fraction of two fractions: 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 . This is the desired form, where both the numerator and the denominator are single fractions.
Final Answer Therefore, the expression with single fractions in the numerator and denominator 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 .
Examples
In electrical engineering, this type of trigonometric manipulation can be used to simplify expressions when analyzing AC circuits. For example, when dealing with impedance calculations involving capacitive and inductive reactances, converting trigonometric functions into simpler fractional forms can make the analysis more straightforward and easier to solve.
To rewrite the expression cos 2 x = s i n 2 x c o s 2 x + 1 s i n 2 x c o s 2 x − 1 , we first rewrite the numerator and denominator with a common denominator. The final result 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 .
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