Prove that
cos(4π+x)+cos(4π−x)=2cosx
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Step-by-Step Solution
Step 1: Apply the sum and difference identities for cosine
We begin by applying the sum and difference identities for cosine to the left-hand side of the equation. The sum identity for cosine is cos(A+B)=cosAcosB−sinAsinB, and the difference identity is cos(A−B)=cosAcosB+sinAsinB. Here, A=4π and B=x.
Step 2: Substitute and expand the terms
Substitute A=4π and B=x into the identities. This expands the left side of the original equation into two parts, one for cos(4π+x) and one for cos(4π−x).
Step 3: Simplify the expression
Notice that the terms −sin4πsinx and +sin4πsinx cancel each other out. This leaves us with two identical terms of cos4πcosx, which can be combined.
Step 4: Substitute the value of cos4π
We know that the value of cos4π (or cos45∘) is 21. Substitute this value into the simplified expression.
Step 5: Rationalize and conclude
To simplify further, we rationalize the denominator by multiplying the numerator and denominator by 2. This results in 222, which simplifies to 2. Thus, the left-hand side simplifies to 2cosx, which is equal to the right-hand side of the original equation.