sin^2x (Power to Double Angle)
The double angle formula for the sine function, written as sin^2x, is a trigonometric identity that represents the square of the sine of twice an angle x
The double angle formula for the sine function, written as sin^2x, is a trigonometric identity that represents the square of the sine of twice an angle x. The formula is derived as follows:
We start with the double angle formula for sine: sin(2x) = 2sin(x)cos(x).
Next, we square both sides of the equation:
(sin(2x))^2 = (2sin(x)cos(x))^2.
Using a trigonometric identity, we expand the equation:
(sin^2(x) + cos^2(x))^2 = (2sin(x)cos(x))^2.
Simplifying the left side using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we have:
1^2 = (2sin(x)cos(x))^2.
Which simplifies to:
1 = 4sin^2(x)cos^2(x).
Finally, dividing both sides of the equation by 4cos^2(x), we get:
sin^2(x) = (1/4)(1/cos^2(x)).
Hence, the double angle formula sin^2x is equal to (1/4)(1/cos^2(x)).
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