Explain how to do this with steps please.
1. Simplify cos(x-y)+cos(x+y)/cosx
I did some of these so far, don't know if it is correct.
Formula: cosxcosy= cos(x+y)+cos(x-y)/2
cos(x-y)+cos(x+y)/cosx
=cosxcosy/2cosx
To simplify the expression cos(x-y)+cos(x+y)/cosx, you can follow these steps:
Step 1: Use the sum-to-product formula for cosine.
The sum-to-product formula states that cos(a) + cos(b) = 2*cos((a+b)/2)*cos((a-b)/2). In this case, let's replace a with (x+y) and b with (x-y):
cos(x-y) + cos(x+y) = 2*cos((x+y+x-y)/2)*cos((x+y-x+y)/2)
= 2*cos(x)*cos(y)
Step 2: Substitute in the simplified expression.
Now, the expression becomes:
2*cos(x)*cos(y) / cos(x)
Step 3: Simplify by canceling out the common factor.
Since there is a common factor of cos(x) in the numerator and denominator, we can cancel them out:
2*cos(x)*cos(y) / cos(x) = 2*cos(y)
Therefore, the simplified expression is 2*cos(y).