Final answer:
The question asks to prove an incorrect version of the cosine subtraction formula.
The correct formula, cos(θ1 - θ2) = cos(θ1)cos(θ2) + sin(θ1)sin(θ2), can be proven by using the cosine of a sum identity and making an angular substitution.
Step-by-step explanation:
The student has asked for a proof of the trigonometric identity cos(θ1 - θ2) = cos(θ1)cos(θ2). There seems to be a typo in the question, as the second θ symbol is missing a numerical subscript. Assuming the equation to prove is cos(θ1 - θ2) = cos(θ1)cos(θ2) + sin(θ1)sin(θ2), which is the correct cosine subtraction formula, here is the proof using the angle sum and difference identities:
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- cos(θ1 - θ2) = cos(θ1)cos(θ2) + sin(θ1)sin(θ2)
This identity can be derived by considering the formulae for cosine of a sum, cos(θ1 + θ2), and then replacing θ2 with -θ2 to find the corresponding formula for cosine of a difference. Using the angle sum identity:
cos(θ1 + θ2) = cos(θ1)cos(θ2) - sin(θ1)sin(θ2)
we can replace θ2 with -θ2 to obtain:
cos(θ1 - θ2) = cos(θ1)cos(-θ2) - sin(θ1)sin(-θ2)
Since cos(-θ) = cos(θ) and sin(-θ) = -sin(θ), we have:
cos(θ1 - θ2) = cos(θ1)cos(θ2) + sin(θ1)sin(θ2)