Question

Express cos 2A in terms of cos 4A..

Answer 1

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cos 2A in Terms of A


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We will learn to express trigonometric function of cos 2A in terms of A. We know if A is a given angle then 2A is known as multiple angles.

How to proof the formula of cos 2A is equals cos2 A - sin2 A?

Or

How to proof the formula of cos 2A is equals 1 - 2 sin2 A?

Or

How to proof the formula of cos 2A is equals 2 cos2 A - 1?

We know that for two real numbers or angles A and B,

cos (A + B) = cos A cos B - sin A sin B






Now, putting B = A on both sides of the above formula we get,

cos (A + A) = cos A cos A - sin A sin A

⇒ cos 2A = cos2 A - sin2 A

⇒ cos 2A = cos2 A - (1 - cos2 A), [since we know that sin2 θ = 1 - cos2 θ]

⇒ cos 2A = cos2 A - 1 + cos2 A,

⇒ cos 2A = 2 cos2 A - 1

⇒ cos 2A = 2 (1 - sin2 A) - 1, [since we know that cos2 θ = 1 - sin2 θ]

⇒ cos 2A = 2 - 2 sin2 A - 1

⇒ cos 2A = 1 - 2 sin2 A

Answer 2

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$\huge{\pink{\underline{\mathfrak{Answer \: is}}}}$

$<b>Answer:</b>$

√(1 + cos 4A)/2

$<b>Step-by-step explanation:</b>$

$<b><u>Theorem:</b></u>$

cos² 2A = (2 cos² 2A)/2

             = (1 - 1 + 2cos²2A)/2

             = [(1 + (2 cos²(2A) - 1)/2]

∴ cos 2A = 2 cos² A - 1

             = [1 + cos2(2A)/2]

             = [1 + cos4A/2]

Now,

We know that cos²(2A) = [1 + cos4A]/2

cos 2A = √(1 + cos 4A)/2

Hope it helps!

$\large{\sf{Regards\: by \: @shreyasingh31}}$


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