To simplify the expression (sinx + sin3x) / (cosx + cos3x), we can use the sum-to-product identities for sine and cosine functions.
First, rewrite the expression as:
sinx/cosx + sin3x/cos3x
Now, apply the sum-to-product identity sin(A) / cos(B) = tan(A-B):
tan(x) + tan(3x)
Now, we can simplify tan(3x) using the triple angle formula for tangent:
tan(3x) = (3tan(x) - tan^3(x)) / (1 - 3tan^2(x))
Therefore, the final simplified expression is:
tan(x) + (3tan(x) - tan^3(x)) / (1 - 3tan^2(x))
To simplify the expression (sinx + sin3x) / (cosx + cos3x), we can use the sum-to-product identities for sine and cosine functions.
First, rewrite the expression as:
sinx/cosx + sin3x/cos3x
Now, apply the sum-to-product identity sin(A) / cos(B) = tan(A-B):
tan(x) + tan(3x)
Now, we can simplify tan(3x) using the triple angle formula for tangent:
tan(3x) = (3tan(x) - tan^3(x)) / (1 - 3tan^2(x))
Therefore, the final simplified expression is:
tan(x) + (3tan(x) - tan^3(x)) / (1 - 3tan^2(x))