Radical ((1-sinx)/(1+sinx))= (1-sinx)/cosx (absolute)
Identity identity
To prove the identity
Radical ((1-sinx)/(1+sinx)) = (1-sinx)/cosx
we can start by simplifying both sides.
Let's begin with the left side of the equation.
Radical ((1-sinx)/(1+sinx))
We can simplify the expression under the radical by multiplying the numerator and denominator by the conjugate of the denominator, which is (1 - sinx).
(1 - sinx) * (1 - sinx) = (1 - sinx)^2 = 1 - 2sinx + sin^2(x)
Now, we can rewrite the expression as:
Radical ((1 - sinx)^2 / (1 + sinx))
Taking the square root of the numerator:
Radical (1 - 2sinx + sin^2(x))
The square root of the denominator remains unchanged:
Radical (1 + sinx)
Now, let's simplify the right side of the equation.
(1 - sinx) / cosx
To simplify the expression, we need to multiply both the numerator and denominator by the conjugate of the denominator, which is cosx.
(1 - sinx) * cosx = cosx - sinx*cosx = cosx - sinx*cos^2(x)/cosx
Simplifying further:
cosx - sinx*cos^2(x)/cosx = cosx - sinx*cos(x) = cosx - sin^2(x)
Now, let's compare the simplified expressions from both sides of the equation:
Radical (1 - 2sinx + sin^2(x)) = cosx - sin^2(x)
To prove the identity, we need to show that both sides of the equation are equal.
Taking the square of the expression on the right side:
(cosx - sin^2(x))^2 = cos^2(x) - 2sin^2(x)*cosx + sin^4(x)
Expanding the expression:
cos^2(x) - 2sin^2(x)*cosx + sin^4(x) = cos^2(x) - 2sin^2(x)*cosx + sin^2(x) * sin^2(x)
Rearranging the terms:
cos^2(x) - 2sin^2(x)*cosx + sin^2(x) * sin^2(x) = cos^2(x) + sin^2(x)*(1 - 2cosx + sin^2(x))
Using the identity sin^2(x) + cos^2(x) = 1, we can simplify further:
cos^2(x) + sin^2(x)*(1 - 2cosx + sin^2(x)) = cos^2(x) + sin^2(x)*(1 - 2cosx + 1)
Simplifying the expression inside the parentheses:
cos^2(x) + sin^2(x)*(1 - 2cosx + 1) = cos^2(x) + sin^2(x)*(2 - 2cosx)
Now, using the identity sin^2(x) = 1 - cos^2(x), we can simplify the expression even further:
cos^2(x) + sin^2(x)*(2 - 2cosx) = cos^2(x) + (1 - cos^2(x))*(2 - 2cosx)
Expanding the expression:
cos^2(x) + (1 - cos^2(x))*(2 - 2cosx) = cos^2(x) + 2 - 2cosx - cos^2(x)*(2 - 2cosx)
Simplifying:
cos^2(x) + 2 - 2cosx - cos^2(x)*(2 - 2cosx) = 2 - 2cosx + 2cosx - 2cos^2(x) = 2 - 2cos^2(x)
Finally, we can see that:
Radical (1 - 2sinx + sin^2(x)) = cosx - sin^2(x) = 2 - 2cos^2(x)
Therefore, the given identity holds true.