Prove that cos2x=(1-tan^2x)/(1+tan^2x)
I can't find any identities that can help me with this. I'm really confused
1 + tan^2 x= (cos^2 x+ sin^2 x)/cos^2 x = 1/cos^2 x
Therefore
=(1-tan^2x)/(1+tan^2x)= cos^2x - sin^2x
= cos 2x
To prove the identity cos(2x) = (1 - tan^2(x))/(1 + tan^2(x)), we can start with the Pythagorean identity for tangent:
tan^2(x) + 1 = sec^2(x)
Now, let's manipulate this equation to find tan^2(x) in terms of sec^2(x):
tan^2(x) = sec^2(x) - 1
Next, we can substitute this expression for tan^2(x) into the original equation:
cos(2x) = (1 - tan^2(x))/(1 + tan^2(x))
= (1 - (sec^2(x) - 1))/(1 + (sec^2(x) - 1))
= (2 - sec^2(x))/(2 + sec^2(x))
Now, we can use the identity:
sec^2(x) = 1 + tan^2(x)
to simplify the expression further:
cos(2x) = (2 - sec^2(x))/(2 + sec^2(x))
= (2 - (1 + tan^2(x)))/(2 + (1 + tan^2(x)))
= (1 - tan^2(x) + 2)/(1 + tan^2(x) + 2)
= 3 - tan^2(x)/(3 + tan^2(x))
= 1 - (tan^2(x)/(1 + tan^2(x)))
Finally, we can use the identity:
1 = sec^2(x)
to simplify the expression even more:
cos(2x) = 1 - (tan^2(x)/(1 + tan^2(x)))
= 1 - (1 - sec^2(x))/(1 + (1 - sec^2(x)))
= 1 - (1 - sec^2(x))/(2 - sec^2(x))
= (sec^2(x) - 1)/(sec^2(x) + 1)
Hence, we have proven that cos(2x) = (1 - tan^2(x))/(1 + tan^2(x)).