can you prove this equation for me?
sin2x X sec2x = 2sinx
by "proving" an equation , we usually mean to show that it is an identity.
but it is NOT an identity.
(I usually just try it with an arbitrary angle)
sin2x/cos2s = 2sinx
tan 2x = 2sinx
let x= 30°
LS = tan 60 = √3
RS = 2sin30 = 2(1/2) = 1
LS ≠ RS
So it is not an identity, (all we need is one exception)
If you are solving for x, then
2sinx cosx (1/cos2x) = 2sinx
2sinxcosx(1/(2cos^2x - 1) = 2sinx
2sinxcosx = 2sinx(2cos2x - 1)
2sinxcosx - 2sinx(2cos^2x - 1) = 0
2sinx [ cosx - 2cos^2x + 1] = 0
2sinx = 0 or 2cos^2x - cosx - 1) = 0
sinx = 0
x = 0, 180°, 360°
2cos^2x - cosx - 1) = 0
(2cosx + 1)(cosx - 1) = 0
cosx = -1/2 or cosx = 1
cosx = -1/2
x = 120°, 240°
cosx = -1
x = 270°
x = 0, 180, 360, 120, 240, 270 °
I gave the angles in degrees, and in the domain between 0 and 360
I assume you know how to change them to radians if necessary.
To prove the equation sin(2x) * sec^2(x) = 2sin(x), we can start by expressing sec^2(x) in terms of sin(x) and cos(x).
Recall that the identity sec^2(x) = 1/cos^2(x) holds true. Now, we need an expression for cos^2(x) in terms of sin(x).
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rearrange it to cos^2(x) = 1 - sin^2(x).
Now we can substitute cos^2(x) in terms of sin(x) in the equation sec^2(x) = 1/cos^2(x):
sec^2(x) = 1 / (1 - sin^2(x))
Next, substitute this expression for sec^2(x) into the original equation:
sin(2x) * (1 / (1 - sin^2(x))) = 2sin(x)
Now let's simplify the left side of the equation:
sin(2x) / (1 - sin^2(x)) = 2sin(x)
To further simplify, we can express sin(2x) as 2sin(x)cos(x) using the double-angle identity for sine:
(2sin(x)cos(x)) / (1 - sin^2(x)) = 2sin(x)
Now, cancel out the common term of sin(x) on both sides of the equation:
(2cos(x)) / (1 - sin^2(x)) = 2
Now, we can multiply both sides of the equation by (1 - sin^2(x)) to eliminate the denominator:
2cos(x) = 2 - 2sin^2(x)
Distribute the 2 on the right side:
2cos(x) = 2 - 2sin^2(x)
Rearrange the terms on the right side:
2cos(x) = 2(1 - sin^2(x))
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, replace cos^2(x) with (1 - sin^2(x)):
2cos(x) = 2(cos^2(x))
Now we have:
2cos(x) = 2cos^2(x)
Divide both sides of the equation by 2:
cos(x) = cos^2(x)
Since cosine is not equal to cosine squared for all values of x, we can conclude that the original equation sin(2x) * sec^2(x) = 2sin(x) cannot be proven true.