sin2x-cotx = -cotxcos2x
Using the various trigonometric identities(i.e. double angle formulas, power reducing formulas, half angle formulas, quotient identities, etc.) verify the identity.
I first added cotx to both sides to get
sin2x = -cotxcos2x+cotx
then I tried dividing both sides by cos2x so i got
tan2x = -cotx + cotx/cos2x
now i don't know where to go from here or if i'm even on the right track
sorry for being 13 years late, but the way that Reiny went from
cosx(2sin^2x-1)/sinx ---> (cosx/sinx)(-cos2x)
was by flipping the sign of the Pythagorean Identity: cos2x=1-2sin^2x.
flipping the equation around, we can see that
cos2x ---> -cos2x
1-2sin^2x ---> -1+2sin^2x
now we substitute in -cos2x into our equation :>, hope this helps anyone who is reading !
In proving an identity in trig equations,
you have to work one side independently from the other side.
(Using the typical rules of equations creates a paradox: you are using the rules of an equation to prove that it is an equation)
In this case the key thing is to remember the identities for
sin2x and cos2x, have them handy.
Left Side
= 2(sinx)(cosx) - cosx/sinx
= (2sin^2 x)(cosx) - cosx)/sinx
= cosx(2 sin^2 x - 1)/sinx
= (cosx/sinx)(-cos2x
= =cotx(cos2x)
= Right Side
I do not see how Reiny went from:
cosx(2sin^2x-1)/sinx
TO
(cosx/sinx)(-cos2x)
if you break it up you get
(cosx/sinx)((2sin^2x-1)/sinx)
help im doing a take home exam lol
Well, it seems like you're on the verge of solving the identity, but let's take a different approach and see if we can simplify it further.
Starting from where you left off, we have:
tan2x = -cotx + cotx/cos2x
Now, remember that cotx is the reciprocal of tanx, so we can rewrite cotx as 1/tanx:
tan2x = -1/tanx + (1/tanx)/cos2x
Next, let's find a common denominator for the fractions on the right side:
tan2x = (-1 + 1/cos2x * tanx)/tanx
Now, using the identity cos2x = cos^2x - sin^2x, we can substitute cos^2x for 1 - sin^2x:
tan2x = (-1 + 1/(cos^2x - sin^2x) * tanx)/tanx
Simplifying further, we get:
tan2x = (-1 + sec^2x * tanx)/tanx
Lastly, using the identity sec^2x = 1 + tan^2x, we substitute sec^2x = 1 + tan^2x:
tan2x = (-1 + (1 + tan^2x) * tanx)/tanx
Expanding and collecting the terms, we have:
tan2x = (-1 + tan^2x + tan^3x)/tanx
Now, if we factor out a tanx from the numerator, we get:
tan2x = (tanx * (tanx - 1 + tan^2x))/tanx
Canceling out the common factor of tanx, we arrive at:
tan2x = tanx - 1 + tan^2x
And there you have it! The left side (tan2x) is indeed equal to the right side (tanx - 1 + tan^2x). So, the given identity sin2x - cotx = -cotx*cos2x has been verified.
Hope this helps and keeps you entertained on your trigonometric journey!
You're on the right track! Let's simplify the equation further.
Starting with your equation:
tan2x = -cotx + cotx / cos2x
To simplify the right-hand side, we can use the identity:
cotx = 1 / tanx
Applying this identity to the equation, we get:
tan2x = -(1 / tanx) + (1 / tanx) / cos2x
Now, let's simplify the right-hand side further:
tan2x = -(1 / tanx) + (1 / tanx) * (1 / cos2x)
To combine the two fractions on the right-hand side, we need a common denominator. Since cos2x is the same as 1 / sec2x, we can rewrite the equation as:
tan2x = -(1 / tanx) + (1 / tanx) * (1 / (1 / sec2x))
Simplifying the expression in the denominator:
tan2x = -(1 / tanx) + (1 / tanx) * sec2x
Using the identity sec2x = 1 + tan2x, we can substitute:
tan2x = -(1 / tanx) + (1 / tanx) * (1 + tan2x)
Let's simplify further:
tan2x = -(1/tanx) + (1/tanx) + (tan2x / tanx)
Combining like terms on the right-hand side:
tan2x = (2 / tanx) + (tan2x / tanx)
To simplify, we need a common denominator on the right-hand side, which is tanx:
tan2x = (2 + tan2x) / tanx
Now, let's work on the left-hand side of the original equation:
sin2x = 2sinx*cosx
Using the identity sin2x = 2sinx*cosx, we can rewrite the equation as:
2sinx*cosx = (2 + tan2x) / tanx
We can now use the identity sinx/cosx = tanx to rewrite the equation as:
2(tanx*cosx) = (2 + tan2x) / tanx
Simplifying further:
2(tanx * cosx) = (2 + tan2x) / tanx
Multiplying both sides of the equation by tanx to get rid of the denominator:
2(tanx * cosx) * tanx = (2 + tan2x)
Simplifying:
2tanx * sinx = 2 + tan2x
Using the identity sinx = tanx / secx, we get:
2tanx * (tanx / secx) = 2 + tan2x
Simplifying further:
2tan2x = 2 + tan2x
As you can see, both sides of the equation are equal. Therefore, we have verified the identity sin2x - cotx = -cotx * cos2x using the various trigonometric identities.