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kasse
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In my book, (cos4x)^2 is written 1+cos8x without referring to any formula. Which trig. identity is used here?
cristo said:Try looking at the identity for cos(2x)
kasse said:You mean cos(2x) = (cosx)^2 - (sinx)^2 ?
JJ420 said:the identity is cos^2x = (1 + cos2x)/2 is it not?
A trigonometric identity is an equation that is true for all values of the variables involved. In other words, it is an identity that holds true for all possible values of the trigonometric functions involved.
The trigonometric identity (cos4x)^2 = 1+cos8x is significant because it is a double angle identity. This means that it can be used to simplify complex trigonometric expressions involving double angles, making calculations easier and more efficient.
You can use this identity to simplify complex trigonometric expressions involving double angles. For example, if you have an expression like cos(8x) + cos(12x), you can use the identity to rewrite it as 2cos(4x)^2. This makes the expression easier to work with and can save time in calculations.
There are several ways to prove the identity (cos4x)^2 = 1+cos8x. One way is to use the double angle formula for cosine, which states that cos(2x) = 2cos^2(x) - 1. By substituting 2x for 4x, we get cos(4x) = 2cos^2(2x) - 1. Then, using the double angle formula again, we get cos(8x) = 2cos^2(4x) - 1. Finally, substituting this into the original identity gives us (cos4x)^2 = 1+cos8x, proving its validity.
Yes, there are several other useful identities that can be derived from (cos4x)^2 = 1+cos8x. For example, you can use it to derive the half angle formula for cosine (cos^2(x/2) = 1/2(1+cos(x)), as well as other double angle identities such as sin(2x) = 2sin(x)cos(x).