1-tanx/1+tanx=1-sin2x/cos2x
see the related questions below.
To solve the equation 1 - tanx / 1 + tanx = 1 - sin(2x) / cos(2x), we can use trigonometric identities to simplify and solve the equation step by step.
First, let's simplify the left-hand side of the equation using the identity:
tan(x) = sin(x) / cos(x)
1 - sin(x) / cos(x) / 1 + sin(x) / cos(x)
Next, we can simplify the right-hand side of the equation using the identity:
sin(2x) = 2sin(x)cos(x)
cos(2x) = cos^2(x) - sin^2(x) = 1 - 2sin^2(x)
Replacing sin(2x) and cos(2x) in the equation:
1 - sin(x)/cos(x) = 1 - (2sin(x)cos(x))/(1 - 2sin^2(x))/(cos(x))
Now, we can start simplifying the equation.
1 - sin(x)/cos(x) = 1 - 2sin(x)cos(x) / (1 - 2sin^2(x))/(cos(x))
To get rid of the complex fraction, we can take the reciprocal of the denominator and multiply it with the numerator:
1 - sin(x)/cos(x) = 1 - 2sin(x)cos(x) * (cos(x)/(1 - 2sin^2(x))
Now, simplifying further:
1 - sin(x)/cos(x) = 1 - 2sin(x)cos(x)cos(x) / (1 - 2sin^2(x))
1 - sin(x)/cos(x) = 1 - 2sin(x)cos^2(x) / (1 - 2sin^2(x))
Now, we can simplify the equation by canceling out common factors:
1 - sin(x)/cos(x) = 1 - 2sin(x)cos^2(x) / (1 - 2sin^2(x))
(1 - sin(x)/cos(x)) * (1 - 2sin^2(x)) = 1 - 2sin(x)cos^2(x)
(1 - sin(x)) * (cos(x)) / (cos(x)) * (1 - 2sin^2(x)) = 1 - 2sin(x)cos^2(x)
(1 - sin(x)) / (1 - 2sin^2(x)) = 1 - 2sin(x)cos^2(x)
Now, we have a simplified equation:
(1 - sin(x)) / (1 - 2sin^2(x)) = 1 - 2sin(x)cos^2(x)
To solve this equation, we can examine the numerator and the denominator separately.
Numerator: (1 - sin(x))
For the numerator be zero, sin(x) must be equal to one.
sin(x) = 1
This occurs when x equals to π/2.
Denominator: (1 - 2sin^2(x))
For the denominator to be zero, sin(x) must be equal to ± 1/√2
sin(x) = ± 1/√2
This occurs when x equals π/4 or 3π/4.
Therefore, the solutions to the equation are x = π/2, π/4, and 3π/4.
(1-tanx)^2/(1-tan^2 x)
= (1-2tanx+tan^2 x)/(1-tan^2 x)
= (1+tan^2 x)/(1-tan^2 x) - 2tanx/(1-tan^2 x)
= sec^2x/(1-tan^2 x) - tan2x
= 1/(cos^2x-sin^2x) - tan2x
= 1/cos2x - sin2x/cos2x
= (1-sin2x)/cos2x
steve can i ask u what happen to the 1-tan^2x in the expression sec^2x/(1-tan^2 x) - tan2x its gone in the next line.