Question

1. use the suitable product-to-sum formula to rewrite the following expression: sin(13x/2)sin(5x/2)

2. Find value of cos(255degrees)cos(105degrees)
is it 1/4?

3. cos(pi/12) - cos(5pi/12)
root3/4?

4. Use the appropriate sum-to-product formula to rewrite the expression sin6x - sin9x
I don't really understand how to do these, but I got -2sin(3x/2)cos(15x/2)..

5. same type of question: rewrite the expression cos4x - cos3x
cosx??

Answer 1

Look at Theorem 1.6 about 3/4 of the way down this British webpage for some lesser know identities

http://personalpages.manchester.ac.uk/staff/david.harris/1C2/1C2Sec2.pdf

so comparing your

sin(13x/2)sin(5x/2)
with
cosA - cosB = -sin((A+B)/2)sin((A-B)/2))
cosB - cosA = sin((x+y)/2)sin((x-y)/2))
and setting
(A+B)/2= 13x/2---> A+B = 13x
(A-B)/2 = 5x/2----> A-B = 5x
adding them
2A = 18x
A = 9x
B = 4x
sin(13x/2)sin(5x/2) = cos(4x) - cos(9x)

2.

Use the appropriate formula from the same section
or
cos(255) = -cos(75)
= -cos(45+30)
= -(cos45cos30 - sin45sin30)
= -( (√2/2)(√3/2) - (√2/2)(1/2)) = -(√6/4 - √2/4)
= √2/4 - √6/4

cos(105) = -cos75 = √2/4 - √6/4
cos(255degrees)cos(105degrees) = (√2-√6)^2/16
= (2 - 2√12 + 6)/16
= (8 - 4√3)/16 = (2 - √3)/4

try the others