Verify that the equation given below is an identity ([tex]\cos 2 x=\cos (x+x)[/tex])
[tex]\cos 2 x=\frac{\cot ^2 x-1}{\cot ^2 x+1}[/tex]
[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{\frac{\cos ^2 x-\sin ^2 x}{\sin ^2 x}}{\frac{1}{\sin ^2 x}}[/tex]
Apply a Pythagorean Identity.
Simplify the complex fraction by multiplying the fraction in the numerator by the reciprocal of the fraction in the denominator
[tex]=\square \text { (Simplify your answer Do not factor ) }[/tex]
Answer (1)
Rewrite the complex fraction as a product: Multiply the numerator by the reciprocal of the denominator.
Simplify by canceling out the sin 2 x terms.
Recognize the resulting expression as the double angle identity for cos 2 x .
The simplified expression is cos 2 x − sin 2 x
Explanation
Analyzing the Problem We are given the equation cos 2 x = c o t 2 x + 1 c o t 2 x − 1 and we want to verify this identity. We are given the intermediate step s i n 2 x 1 s i n 2 x c o s 2 x − s i n 2 x . Our goal is to simplify this complex fraction.
Multiplying by the Reciprocal To simplify the complex fraction s i n 2 x 1 s i n 2 x c o s 2 x − s i n 2 x , we multiply the numerator by the reciprocal of the denominator. The reciprocal of s i n 2 x 1 is sin 2 x . So we have: sin 2 x cos 2 x − sin 2 x × sin 2 x
Simplifying the Expression Now we simplify the expression by canceling out the sin 2 x terms: sin 2 x cos 2 x − sin 2 x × sin 2 x = cos 2 x − sin 2 x
Applying the Double Angle Identity We recognize that cos 2 x − sin 2 x is the double angle identity for cos 2 x . Therefore, the simplified expression is cos 2 x .
Final Answer Thus, the simplified form of the given expression is cos 2 x − sin 2 x , which is equal to cos 2 x .
Examples
Understanding trigonometric identities like the one we just verified is crucial in fields like physics and engineering. For example, when analyzing the motion of a pendulum or the behavior of alternating current in an electrical circuit, trigonometric functions and their identities are used to simplify complex equations and make predictions. In essence, mastering these identities allows engineers and physicists to model and understand oscillatory phenomena more effectively.
