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What is the formula of cos4x?

Answer
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Hint: Here in this question, we have to find the formula of given trigonometric function. To solve this, First rewrite the given angle in the form of addition or difference, then the standard trigonometric formula cosine sum i.e., cos(A+B)or cosine difference i.e., cos(AB) identity defined as cosA.cosBsinA.sinB and cosA.cosB+sinA.sinBusing one of these we get required value.

Complete step-by-step solution:
To evaluate the given question by using a formula of cosine addition defined as the cosine addition formula calculates the cosine of an angle that is either the sum or difference of two other angles. It arises from the law of cosines and the distance formula. By using the cosine addition formula, the cosine of both the sum and difference of two angles can be found with the two angles' sines and cosines.
Consider the given function
cos4x-------(1)
It can be rewritten as
cos(2x+2x)
Now, by using a cosine sum identity: cos(A+B)=cosA.cosBsinA.sinB
Here, A=2x and B=2x on substituting, we have
 cos(2x+2x)=cos2x.cos2xsin2x.sin2x
cos22xsin22x
As by the standard trigonometric identity: sin2θ+cos2θ=1 sin2θ=1cos2θ, then on simplification we have
cos22x(1cos22x)
cos22x1+cos22x
On simplification, we have
2cos22x1
Or it can be written as
2(cos(x+x))21
Again, by the cosine sum identity we have
2(cosxcosxsinxsinx)21
2(cos2xsin2x)21
Again, by the trigonometric identity sin2θ=1cos2θ, we have
2(cos2x(1cos2x))21
2(2cos2x1)21
Now using a algebraic identity (ab)2=a2+b22ab
Here a=2cos2x and b=1, then
2((2cos2x)2+122(2cos2x)(1))1
2(4cos4x+14cos2x)1
On simplification, we have
8cos4x+28cos2x1
8cos4x8cos2x+1

Hence, the formula cos4x=8cos4x8cos2x+1.

Note: This question also can be solved by using a double angle formula cos2x=2cos2x1 and further simplification will be the same in the above method. Since they have mentioned to solve the given function by using the cosine sum or difference identity, for this we have standard formula and also known the trigonometric identities, double and half angle formulas.
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