Expert: Paul Klarreich - 11/25/2009

Question
prove  cos7A/cosA=2cos6A-2cos4A+2cos2A-1

Answer
Questioner: dolly
Country: India
Category: Advanced Math
Private: No
Subject: trignometry
Question: prove  cos7A/cosA=2cos6A-2cos4A+2cos2A-1
............................

cos(7a)
-------  = 2 cos 6A - 2 cos 4A + 2 cos2A - 1
cos a

First cross-multiply.  (Technically illegal, but you can get around it.)
(I am being a bit sloppy with the A's and a's, but,.....)

cos7A = 2 cos 6A cos a - 2 cos4A cos a + 2 cos2A cos a - cos a

Now use the 'PRODUCT-TO-SUM' identities you find at:

en.wikipedia.org/wiki/List_of_trigonometric_identities

to write these:

2 cos 6A cos a = cos 7a + cos 5a
2 cos 4A cos a = cos 5a + cos 3a
2 cos 2a cos a = cos 3a + cos a

Plug them in:

cos7A = [cos 7a + cos 5a] - [cos 5a + cos 3a] + [cos 3a + cos a] - cos a

Remove those brackets carefully and you are home.