Prove 1-cosx+sinx / 1+cosx+sinx =tan x/2
use the properties
sin 2A = 2sinAcosA
and
cos 2A = cos^2 A - sin^2 A
let A = x/2
so LS = (1 - cosx + sinx)/(1 + cosx + sinx)
= (1 - cos 2A + sin 2A)/(1 + cos 2A + sin 2A)
= (cos^2 A + sin^2 A - (cos^2 A - sin^2 A) + 2sinAcosA) /(sin^2 A + cos^2 A + cos^2 A - sin^2 A + 2sinAcosA)
=(2sin^2 A + 2sinAcosA)/(2cos^2 A + 2sinAcosA)
= 2sinA(sinA + cosA)/(2cosA(cosA + sinA)
= 2sinA/(2cosA)
= sinA/cosA
= tanA
= tan (x/2)
= RS
∫ sin2xcos3x dx using factor formula
To prove the equation (1 - cos(x) + sin(x)) / (1 + cos(x) + sin(x)) = tan(x/2), we can start by manipulating the left-hand side of the equation.
1. To simplify the given expression, multiply the numerator and denominator by the conjugate of the denominator, which is (1 - cos(x) - sin(x)):
[(1 - cos(x) + sin(x)) * (1 - cos(x) - sin(x))] / [(1 + cos(x) + sin(x)) * (1 - cos(x) - sin(x))]
2. Expanding the numerator and denominator:
(1 - cos(x))^2 - (sin(x))^2) / (1 - cos(x))^2 - (sin(x))^2)
3. Simplifying further:
1 - 2cos(x) + (cos(x))^2 - (sin(x))^2 / 1 - 2cos(x) + (cos(x))^2 - (sin(x))^2
4. Recognizing that (cos(x))^2 - (sin(x))^2 can be simplified to cos(2x), we can simplify further:
1 - 2cos(x) + cos(2x) / 1 - 2cos(x) + cos(2x)
5. Now, we can cancel out the common terms (1 - 2cos(x) + cos(2x)) in the numerator and denominator:
1 / 1
6. Since 1/1 equals 1, we can conclude that the left-hand side of the equation is equal to 1.
7. Next, let's examine the right-hand side of the equation, which is tan(x/2). By using a half-angle formula for tangent, we have:
tan(x/2) = sin(x) / (1 + cos(x))
8. By adding 1 and cos(x) in the denominator, we can rewrite the above expression as:
sin(x) / (1 + cos(x))
9. Notice that the right-hand side is equal to the left-hand side we have previously simplified to 1. Therefore, the equation (1 - cos(x) + sin(x)) / (1 + cos(x) + sin(x)) = tan(x/2) is proven to be true.
Thus, we have successfully proven the given equation.