Show that 1-cos2A/Cos^2*A = tan^2*A
1-cos2A/Cos^2*A =
[Cos^2(A) - Cos(2A)]/Cos^2(A).
Substitute:
Cos(2A) = 2Cos^2(A) - 1:
[1 - Cos^2(A)]/Cos^2(A)=
Sin^2(A)/Cos^2(A) = tan^2(A)
To prove that 1 - cos^2(2A) / cos^2(A) is equal to tan^2(A), we can start by simplifying the left-hand side of the equation.
1 - cos^2(2A) / cos^2(A)
Now, let's use the identity cos(2A) = 2cos^2(A) - 1. By substituting this identity into the equation, we get:
1 - (2cos^2(A) - 1)^2 / cos^2(A)
Expanding the square and simplifying the numerator:
1 - (4cos^4(A) - 4cos^2(A) + 1) / cos^2(A)
Now, let's combine like terms in the numerator:
1 - 4cos^4(A) + 4cos^2(A) - 1 / cos^2(A)
The 1 and -1 terms cancel out:
-4cos^4(A) + 4cos^2(A) / cos^2(A)
Factoring out a common factor of 4cos^2(A):
4cos^2(A) * (-cos^2(A) + 1) / cos^2(A)
The cos^2(A) terms also cancel out:
4(-cos^2(A) + 1)
Simplifying further:
-4cos^2(A) + 4
Finally, we can express this equation in terms of tan(A). Recall that tan(A) = sin(A) / cos(A):
-4cos^2(A) + 4 = -4(1 - sin^2(A)) + 4 = 4sin^2(A)
Now we have:
4sin^2(A) = tan^2(A)
Therefore, we have shown that 1 - cos^2(2A) / cos^2(A) is equal to tan^2(A).