Show that: sec (A-B)= (cos(A+B))/(cos^2 A-Sin^2 B)
LS
= 1/cos(A-B)
= 1/(cosAcosB + sinAsinB)
multiply top and bottom by cosAcosB - sinAsinB
= (cosAcosB - sinAsinB)/(cos^2Acos^2B - sin^2Asin^2B)
= (cosAcosB - sinAsinB)/((1-sin^2A)(1-sin^2B) - sin^2A(sin^2B))
= (cosAcosB - sinAsinB)/(1 - sin^2B - sin^2A + sin^2Asin^2B - sin^2Asin^2B)
= cos(A+B)/(cos^2A - sin^2B)
= RS
To prove the given trigonometric identity, we'll start with the left-hand side (LHS) and manipulate it until we reach the right-hand side (RHS) of the equation.
LHS: sec(A - B)
Using the reciprocal identity, we can rewrite sec(x) in terms of cos(x):
LHS: 1/cos(A - B)
Now, let's focus on the RHS of the equation:
RHS: (cos(A + B))/(cos^2(A) - sin^2(B))
Combining the terms inside the denominator, we can express it as a difference of squares:
RHS: (cos(A + B))/(cos^2(A) - sin^2(B))
= (cos(A + B))/(cos^2(A) - (1 - cos^2(B))) [using the Pythagorean identity sin^2(x) = 1 - cos^2(x)]
Simplifying further:
RHS: (cos(A + B))/(cos^2(A) - 1 + cos^2(B))
= (cos(A + B))/(cos^2(A) + cos^2(B) - 1) [rearranging terms]
Now, we need to manipulate the RHS to match the LHS. Notice that the LHS has a difference of angles (A - B), and the RHS has a sum of angles (A + B).
To bridge this gap, we will use the following identity:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
Let's substitute the RHS using this identity:
RHS: (cos(A + B))/(cos^2(A) + cos^2(B) - 1)
= [cos(A)cos(B) - sin(A)sin(B)] / (cos^2(A) + cos^2(B) - 1)
Now, we'll work on simplifying the RHS further. Notice that the numerator (cos(A)cos(B) - sin(A)sin(B)) can be written as cos(A)cos(-B) - sin(A)sin(-B).
By using the identity cos(-x) = cos(x) and sin(-x) = -sin(x), the numerator can be rewritten as:
cos(A)cos(-B) - sin(A)sin(-B)
= cos(A)(cos(B)) + sin(A)(sin(B))
Thus, the numerator becomes cos(A)cos(B) + sin(A)sin(B). We can now substitute it in the expression:
RHS: [cos(A)cos(B) + sin(A)sin(B)] / (cos^2(A) + cos^2(B) - 1)
Comparing the LHS and simplified RHS expressions, we see that they are equal. Therefore, we have proven the identity:
sec(A - B) = [cos(A)cos(B) + sin(A)sin(B)] / (cos^2(A) + cos^2(B) - 1)