How to solve this equation

Question

cos(x).cos(2x)cos(4x)cos(8x)= sin16x/16sinx prove this !

Answer 1

I'm going to start with the right side and transform it into the left side

sin(16x)

16sin(x)

 

I'm going to ignore the denominator for a few minutes and come back to it.

 

The strategy for this problem is to expand the sin(16x) multiple times using the double angle formula.  We know that sin(2x) = 2sin(x)cos(x).  Therefore,

 

sin(16x) = 2sin(8x)cos(8x) = 2cos(8x) * sin(8x)

 

Expand the sin(8x) using the double angle formula again (I italicized the parts of the equation that are changing due to the expansion to help you follow the math):

sin(16x) = 2cos(8x) * sin(8x) = 2cos(8x) * [2sin(4x)cos(4x)] = 2*2*cos(8x)cos(4x)*sin(4x)

= 4cos(8x)cos(4x)sin(4x)

 

Expand the sin(4x) using the double angle formula:

sin(16x) = 4cos(8x)cos(4x)*[sin(4x)] = 4cos(8x)cos(4x)*[2sin(2x)cos(2x)]  = 4*2*cos(8x)cos(4x)cos(2x)sin(2x)

=8*cos(8x)cos(4x)cos(2x)sin(2x)

 

Expand the sin(2x) term using the double angle formula:

sin(16x) = 8*cos(8x)cos(4x)cos(2x)* [sin(2x)] = 8*cos(8x)cos(4x)cos(2x)*[2*sin(x)cos(x)] = 8*2*cos(8x)cos(4x)cos(2x)cos(x)sin(x) = 16cos(8x)cos(4x)cos(2x)cos(x)sin(x)

 

Ok, now we've expanded sin(16x) completely and we know that:

sin(16x) = 16cos(8x)cos(4x)cos(2x)cos(x)sin(x)

 

Now we can substitute that back into the original fraction:

sin(16x)
16sin(x)

 

= 16cos(8x)cos(4x)cos(2x)cos(x)sin(x)

                         16sin(x)

Cancel the 16's and sin(x) in both the top and bottom of the fraction:

 

= 16cos(8x)cos(4x)cos(2x)cos(x)sin(x)
                        16sin(x)

 

= cos(8x)cos(4x)cos(2x)cos(x)

 

And you're done :)