(sinx-cosx+1)/(sinx+cosx-1)=(sinx+1)/(cosx)
you have
(sinx-(cosx-1))/(sinx+(cosx-1))
multiply top and bottom by
(sinx-(cosx-1)) and you have
(sinx-(cosx-1))^2/(sin^2x - (cosx-1)^2)
(sin^2x - 2sinx(cosx-1) + (cosx-1)^2)
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(sin^2x - cos^2x + 2cosx - 1)
(sin^2x - 2sinx*cosx + 2sinx + cos^2x - 2cosx + 1)
----------------------------------------
(sin^2x - cos^2x + 2cosx - 1)
2(1 - sinx*cosx + sinx - cosx)
---------------------------------
2(cosx - cos^2x)
((1+sinx) - cosx(1+sinx))
------------------------------
cosx(1-cosx)
(1+sinx)(1-cosx)
-------------------
cosx(1-cosx)
(1+sinx)/cosx
To solve the equation (sinx - cosx + 1) / (sinx + cosx - 1) = (sinx + 1) / cosx, we can follow these steps:
Step 1: Simplify both sides of the equation
Start by expanding the denominators on both sides:
(sin x - cos x + 1) * cos x = (sin x + 1) * (sin x + cos x - 1)
This simplifies to:
sin x * cos x - cos^2 x + cos x = sin x * sin x + sin x * cos x - sin x + sin x * cos x + cos x^2 - cos x
Simplifying further, we get:
sin x * cos x - cos^2 x + cos x = sin^2 x + sin x * cos x - sin x + sin x * cos x + cos x^2 - cos x
Step 2: Collect like terms on both sides
sin x * cos x terms on the left side and sin^2 x terms on the right side can be canceled out:
-cos^2 x + cos x = sin^2 x + cos x^2
Step 3: Simplify the equations
Rearrange the equation to put all terms on one side:
cos^2 x + sin^2 x + cos x^2 - cos x - cos x + cos^2 x = 0
After combining like terms, we get:
2cos^2 x - 2cos x = 0
Step 4: Factor out common terms
Factor out 2cos x:
2cos x (cos x - 1) = 0
Step 5: Solve for cos x
Set each factor equal to zero and solve two separate equations:
cos x = 0 or cos x - 1 = 0
When cos x = 0, we can find the solutions for x by using the inverse cosine function:
x = arccos(0) + 2πn, where n is an integer
x = π/2 + 2πn, where n is an integer
When cos x - 1 = 0, we solve for x using the inverse cosine function:
cos x - 1 = 0 becomes cos x = 1
x = arccos(1) + 2πn, where n is an integer
x = 2πn, where n is an integer
Therefore, the solutions to the equation (sinx - cosx + 1) / (sinx + cosx - 1) = (sinx + 1) / cosx are:
x = π/2 + 2πn, x = 2πn, where n is an integer.