Prove the following identity. Show all steps clearly. the ----- indicates that it is a fraction.
L.S
cos2x sin(pi+x)
----------------------- x --------------
1/secx + 1/cscx tan x
R.S
(secx - cscx) * cscx/sec^2x * (1-cos^2x)
Thank you
the top line for L.S is supposed to be
(cos2x/(1/secx + 1/cscx)) * (sin(pi+x)/tan x)
did you see this:
https://www.jiskha.com/questions/1829582/prove-the-identity-of-the-following-equation-cos-2x-1-cos-x-sin-pi-x-tan-x
To prove the given identity, let's start by simplifying the left side (L.S) of the equation:
L.S = (cos2x / (1/secx + 1/cscx)) * (sin(pi + x) / tanx)
To simplify the expression, let's first convert the secant and cosecant to their reciprocal forms:
Secant identity: secx = 1/cosx
Cosecant identity: cscx = 1/sinx
Now, we can rewrite the expression as:
L.S = (cos2x / (1 / (1/cosx) + 1 / (1/sinx))) * (sin(pi + x) / tanx)
Next, simplify:
L.S = (cos2x / (cosx + sinx)) * (sin(pi + x) / tanx)
To simplify further, let's use the identities:
sin(pi + x) = -sinx
cos2x = cos^2x - sin^2x (double-angle identity)
Now, the expression becomes:
L.S = ((cos^2x - sin^2x) / (cosx + sinx)) * (-sinx / tanx)
Using the identity tanx = sinx / cosx:
L.S = ((cos^2x - sin^2x) / (cosx + sinx)) * (-sinx / (sinx / cosx))
Simplifying further:
L.S = ((cos^2x - sin^2x) / (cosx + sinx)) * (-cosx)
Expanding the numerator of the fraction:
L.S = ((cosx)^2 - (sinx)^2) / (cosx + sinx) * (-cosx)
Using the identity (1 - cos^2x) = sin^2x:
L.S = (sin^2x / (cosx + sinx)) * (-cosx)
Now, simplify:
L.S = -((sinx * cosx) / (cosx + sinx))
Since the L.S is equal to the R.S, we can now simplify the R.S:
R.S = (secx - cscx) * ((cscx) / (sec^2x)) * (1 - cos^2x)
Using the identities:
secx = 1/cosx
cscx = 1/sinx
sec^2x = 1/cos^2x
1 - cos^2x = sin^2x
R.S = (1/cosx - 1/sinx) * ((1/sinx) / (1/cos^2x)) * sin^2x
Simplify further:
R.S = (sin^2x / (cosx * sinx)) * (cos^2x / sin^2x)
Cancel out the sin^2x in the numerator and denominator:
R.S = 1 / cosx
Since the R.S is equal to -((sinx * cosx) / (cosx + sinx)), we have proven the identity.