Need help solving problems using the Trigonometric Identities: 8cos^4 thet=cos4theta+4cos2theta+3
I am a bit unsure how to solve it. I am aware that the first step I must take is to simplify the problem and then use the Product to Sum formula
Nasty one!
let's start on the right side.
I will work on each term to make typing easier.
recall that cos 2A = 2cos^2 A - 1
I will use x for theta
cos 4x = 2cos^2 2x - 1
= 2(cos2x)(cos2x) - 1
=2[(2cos^2 x - 1)(2cos^2 x - 1)] - 1.
=2[(4cos^4 x - 4cos^2 x + 1) – 1
=8cos^4 x – 8cos^2 x + 1
4cos 2x = 4(2cos^2 x – 1)
= 8cos^2 x – 4
Now putting it all together:
R.S.
= cos 4x + 4cos 2x + 3
= 8cos^4 x – 8cos^2 x + 1 + 8cos^2 x – 4 + 3
= 8cos^4 x
= Left Side
!!!!!!!WOW!!!!!!
To solve the equation using trigonometric identities, we can start by simplifying the equation on the right side.
The first term, cos 4theta, can be simplified using the identity cos 4theta = 8cos^4 theta - 8cos^2 theta + 1.
The second term, 4cos 2theta, can be simplified using the identity cos 2theta = 2cos^2 theta - 1. So, 4cos 2theta = 4(2cos^2 theta - 1) = 8cos^2 theta - 4.
Now let's put the simplified terms back into the equation:
8cos^4 theta - 8cos^2 theta + 1 + 8cos^2 theta - 4 + 3
Simplifying further, the -8cos^2 theta and +8cos^2 theta cancel each other out:
8cos^4 theta - 8cos^2 theta + 8cos^2 theta - 8 + 3
Simplifying again:
8cos^4 theta - 8 + 3
Finally, we have:
8cos^4 theta - 5
So, the equation becomes:
8cos^4 theta = 8cos^4 theta - 5
This equation is not possible to solve as both sides are equal to each other. Therefore, there is no solution.