can you simplify cos(15degrees) using compound angel identity?

Answer 1

Yes, we can simplify cos(15 degrees) using the compound angle identity for cosine. According to the compound angle identity:

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

In this case, we have cos(15 degrees), which can be written as:

cos(15 degrees) = cos(45 degrees - 30 degrees)

Using the compound angle identity, we can rewrite it as:

cos(15 degrees) = cos(45 degrees)cos(30 degrees) - sin(45 degrees)sin(30 degrees)

Now, let's evaluate the trigonometric values:

cos(45 degrees) = sqrt(2)/2
cos(30 degrees) = sqrt(3)/2
sin(45 degrees) = sqrt(2)/2
sin(30 degrees) = 1/2

Substituting these values back into the equation, we get:

cos(15 degrees) = (sqrt(2)/2)(sqrt(3)/2) - (sqrt(2)/2)(1/2)

Simplifying further:

cos(15 degrees) = sqrt(6)/4 - sqrt(2)/4

So, cos(15 degrees) can be simplified to (sqrt(6)-sqrt(2))/4.