Establish the identity.
sinx + cosx/sinx - cosx = 1+2sinxcosx/2sin^2x-1
Do you mean
(sinx + cosx)/(sinx - cosx) = (1+2sinxcosx)/(2sin^2x-1 ) ?????????
If so then
[(sinx + cosx)/(sinx - cosx)][(sinx+cosx)/(sinx + cosx)]
= [sinx + cosx]^2/[sin^2x-cos^2x]
= [1 + 2 sin x cos x] / [sin^2x - (1-sin^2x)]
= [1 + 2 sin x cos x] / [2 sin^2 x - 1]
To establish the identity, we need to simplify both sides of the equation and show that they are equal.
First, let's simplify the left side of the equation:
sinx + cosx / sinx - cosx
To simplify this expression, we can multiply both the numerator and the denominator by the conjugate of the denominator, which is sinx + cosx:
(sin x + cos x) * (sin x + cos x) / (sin x - cos x) * (sin x + cos x)
Expanding the denominator using the distributive property, we have:
(sin^2 x + 2sin x cos x + cos^2 x) / (sin^2 x - cos^2 x)
Applying the Pythagorean identity sin^2 x + cos^2 x = 1, we can simplify this expression further:
(1 + 2sin x cos x) / (sin^2 x - cos^2 x)
Now, let's simplify the right side of the equation:
1 + 2sin x cos x / (2sin^2 x - 1)
Since there is only one term on the right side of the equation, we don't need to do any additional simplification.
Now, we can see that both sides of the equation are equal:
(1 + 2sin x cos x) / (sin^2 x - cos^2 x) = 1 + 2sin x cos x / (2sin^2 x - 1)
Hence, the identity has been established.
To establish the identity:
1. Start with the given expression: sin(x) + cos(x) / sin(x) - cos(x) = 1 + 2sin(x)cos(x) / 2sin(x)^2 - 1
2. Multiply the numerator and denominator of the fraction on the right side by the conjugate of the denominator, which is sin(x) + cos(x). This is done to eliminate the difference of squares in the denominator.
(1 + 2sin(x)cos(x)) * (sin(x) + cos(x)) / ((sin(x) + cos(x))(sin(x) - cos(x)))
3. Distribute the numerator on the right side:
= (sin(x) + cos(x)) + 2sin(x)cos(x)(sin(x) + cos(x)) / ((sin(x) + cos(x))(sin(x) - cos(x)))
4. Cancel out the common factor of sin(x) + cos(x):
= 1 + 2sin(x)cos(x) / (sin(x) - cos(x))
5. To simplify further, we can use the identities: sin(2x) = 2sin(x)cos(x) and cos^2(x) + sin^2(x) = 1.
= 1 + sin(2x) / (sin(x) - cos(x))
Therefore, the simplified form of the given identity is: 1 + sin(2x) / (sin(x) - cos(x)).