Prove:
sin^2(x/2) = csc^2x - cot^2x / 2csc^2(x) + 2csc(x)cot(x)
On the right, factor the numberator as a difference of two perfect squares. In the denominator, factor out 2cscx.
You ought to prodeed rather quickly to the proof.
To prove the equation sin^2(x/2) = (csc^2x - cot^2x) / (2csc^2(x) + 2csc(x)cot(x)), we will start by simplifying the right-hand side using some trigonometric identities.
Step 1: Factor the numerator as a difference of two perfect squares
(csc^2x - cot^2x) can be factored as ((cscx + cotx)(cscx - cotx)) using the difference of squares formula.
Step 2: Factor out 2cscx from the denominator
In the denominator, we can factor out 2cscx from each term:
2csc^2(x) + 2csc(x)cot(x) = 2cscx(cscx + cotx)
Step 3: Simplify the right-hand side expression
Now, substituting the factored forms from steps 1 and 2 into the equation, we get:
(sin^2(x/2)) = ((cscx + cotx)(cscx - cotx)) / (2cscx(cscx + cotx))
Step 4: Cancel out common terms
Notice that (cscx + cotx) cancels out in the numerator and denominator, leaving us with:
(sin^2(x/2)) = (cscx - cotx) / (2cscx)
Step 5: Using the reciprocal identity
The reciprocal identity states that cscx = 1 / sinx, and cotx = cosx / sinx. Applying this identity to the equation, we have:
(sin^2(x/2)) = (1/sinx - cosx/sinx) / (2(1/sinx))
Simplifying further, we get:
(sin^2(x/2)) = (1 - cosx) / (2sinx)
Step 6: Using the Pythagorean identity
The Pythagorean identity states that sin^2x + cos^2x = 1. Rearranging this equation, we have:
sin^2x = 1 - cos^2x. Substituting this into the equation, we get:
(sin^2(x/2)) = (1 - cosx) / (2sinx)
Step 7: Using the half-angle identity
The half-angle identity states that sin(x/2) = ±sqrt((1 - cosx)/2). Taking the positive root, we have:
sin(x/2) = sqrt((1 - cosx)/2)
Step 8: Squaring both sides
To get rid of the square root, we square both sides of the equation:
sin^2(x/2) = (sqrt((1 - cosx)/2))^2
Step 9: Simplifying
When we square the square root, the two operations cancel out, resulting in:
sin^2(x/2) = (1 - cosx) / 2
Therefore, we have shown that sin^2(x/2) = (1 - cosx) / 2, which is equivalent to the original equation.
Note: This proof assumes that all trigonometric functions are defined and well-behaved in the given domain.