can you help me with this?
cos2A=cossquareA-sinsquareA/cossquareA+sinsquareA=cossquareA(1-tansquareA)/cossquareA(1+tansquareA)
I encountered this situation while solving trignometric problem.can you tell how does this come?
Certainly! The equation you provided is an identity known as the double angle identity for cosine, which states that:
cos(2A) = cos^2(A) - sin^2(A)
To understand how this identity is derived, we can start with the sum-to-product identity for cosine:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
Now, let's set A = B. We have:
cos(2A) = cos(A + A) = cos(A)cos(A) - sin(A)sin(A)
Using the fact that cos(A)cos(A) = cos^2(A) and sin(A)sin(A) = sin^2(A), we get:
cos(2A) = cos^2(A) - sin^2(A)
Therefore, we have the double angle identity for cosine:
cos(2A) = cos^2(A) - sin^2(A)
This identity can also be expressed in terms of sine and cosine by using the Pythagorean identity:
sin^2(A) + cos^2(A) = 1
By rearranging this identity, we get:
sin^2(A) = 1 - cos^2(A)
Substituting this into the double angle identity for cosine, we have:
cos(2A) = cos^2(A) - (1 - cos^2(A))
Simplifying further, we get:
cos(2A) = 2cos^2(A) - 1
So, the double angle identity for cosine can be written in this alternative form as well:
cos(2A) = 2cos^2(A) - 1
I hope this explanation clarifies how the given equation is derived from the double angle identity for cosine.