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We will learn step-by-step the proof of compound angle formula cos^2 α - sin^2 β. We need to take the help of the formula of cos (α + β) and cos (α - β) to proof the formula of cos^2 α - sin^2 β for any positive or negative values of α and β.
Prove that: cos (α + β) cos (α - β) = cos2 α - sin2 β = cos2 β - sin2 α.
Proof: cos (α + β) cos (α - β)
= (cos α cos β - sin α sin β) (cos α cos β + sin α sin β)
= (cos α cos β)2 - (sin α sin β)2
= cos2 α cos2 β - sin2 α sin2 β
= cos2 α (1 - sin2 β) - (1 - cos2 α) sin2 β, [since we know, cos2 θ = 1 - sin2 θ]
= cos2 α - cos2 α sin2 β - sin2 β + cos2 α sin2 β
= cos2 α - sin2 β
= 1 - sin2 α - (1 - cos2 β), [since we know, cos2 θ = 1 - sin2 θ and sin2 θ = 1 - cos2 θ]
= 1 - sin2 α - 1 + cos2 β
= cos2 β - sin2 α Proved
Therefore, cos (α + β) cos (α - β) = cos2 α - sin2 β = cos2 β - sin2 α
Solved examples using the proof of compound angle formula cos2α - sin2 β:
1. Prove that: cos2 2x - sin2 x = cos x cos 3x.
Solution:
L.H.S. = cos2 2x - sin2 x
= cos (2x + x) cos (2x - x), [since we know cos2 α - sin2 β = cos (α + β) cos (α - β)]
= cos 3x cos x = R.H.S. Proved
2. Find the value of cos2 (π8 - θ2) - sin2 (π8 + θ2).
Solution:
cos2 (π8 - θ2) - sin2 (π8 + θ2)
= cos {(π8 - θ2) + (π8 + θ2)} cos {(π8 - θ2) - (π8 + θ2)},
[since we know, cos2 α - sin2 β = cos (α + β)
cos (α - β)]
= cos {π8 - θ2 + π8 + θ2} cos {π8 - θ2 - π8 - θ2}
= cos {π8 + π8} cos {- θ2 - θ2}
= cos π4 cos (- θ)
= cos π4 cos θ, [since we know, cos (- θ) = cos θ)
= 1√2 ∙ cos θ [we know, cos π4 = 1√2]
3. Evaluate: cos2 (π4 + x) - sin2 (π4 - x)
Solution:
cos2 (π4 + x) - sin2 (π4 - x)
= cos {(π4 + x) + (π4 - x)} cos {(π4 + x) - (π4 - x)}, [since we know, cos2 β - sin2 α = cos (α + β)
cos (α - β)]
= cos {π4 + x + π4 - x} cos {π4 + x - π4 + x}
= cos {π4+π4} cos {x + x}
= cos π4 cos 2x
= 0 ∙ cos 2x, [Since we know, cos π4 = 0]
= 0
11 and 12 Grade Math
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