Simplify. [tex]\frac{2}{\sec ^2 \frac{x}{2}}-1[/tex]
Answer (1)
Rewrite the expression using the identity sec θ = c o s θ 1 .
Simplify the expression to 2 cos 2 2 x − 1 .
Use the double angle identity cos x = 2 cos 2 2 x − 1 to further simplify the expression.
The simplified expression is cos ( x ) .
Explanation
Problem Analysis We are asked to simplify the expression s e c 2 2 x 2 − 1 .
Rewriting the Expression We know that sec θ = c o s θ 1 . Therefore, sec 2 2 x = c o s 2 2 x 1 . Substituting this into the given expression, we have sec 2 2 x 2 − 1 = c o s 2 2 x 1 2 − 1 = 2 cos 2 2 x − 1.
Applying the Double Angle Identity We can use the double angle identity for cosine, which states that cos ( 2 θ ) = 2 cos 2 ( θ ) − 1 . In our case, we have 2 cos 2 2 x − 1 , which matches the right-hand side of the double angle identity with θ = 2 x . Therefore, 2 cos 2 2 x − 1 = cos ( 2 ⋅ 2 x ) = cos ( x ) .
Final Answer Thus, the simplified expression is cos ( x ) .
Examples
In electrical engineering, when analyzing AC circuits, you often encounter trigonometric functions. Simplifying expressions like the one above can help in determining the phase relationships between voltage and current. For example, if you have an expression involving sec 2 ( 2 x ) in the context of power factor correction, simplifying it to cos ( x ) can make calculations easier and provide a clearer understanding of the circuit's behavior.
