Final answer:
The Double-Angle formula can indeed be used to find sin 22.5° by applying the half-angle formula. An example is provided to demonstrate the use of the formula. Additionally, when dividing the Pythagorean Identity by sin²x, it simplifies to 1 + cot²x = csc²x.
Step-by-step explanation:
Yes, the Double-Angle formula can be used to find sin 22.5°. This is because 22.5° is half of 45°, and by using the half-angle formula for sine, we can find the value. For instance, sin(45°/2) = ±√((1 - cos 45°)/2). Since 22.5° is in the first quadrant, where sine is positive, we only consider the positive root.
Example problem using the half-angle formula:
Find sin 22.5° using the half-angle formula.
Solution:
sin(22.5°) = sin(45°/2) = √((1 - cos(45°))/2) = √((1 - √2/2)/2)
When we divide the Pythagorean Identity sin²x + cos²x = 1 by sin²x, we get the following simplifications:
1 + (cos²x/sin²x) = 1/sin²x
1 + cot²x = csc²x