Maharashtra State BoardHSC Science (Electronics) 11th
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Prove the following: In ∆ABC, ∠C = 2π3, then prove that cos2A + cos2B − cos A cos B = 34 - Mathematics and Statistics

Sum

Prove the following:

In ∆ABC, ∠C = `(2pi)/3`, then prove that cos2A + cos2B − cos A cos B = `3/4`

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Solution

In ∆ABC, A + B + C = π, where ∠C = `(2pi)/3`

∴ A + B = π  – C = `pi/3`   ...(1)

L.H.S. = cos2A + cos2B – cos A cos B

= cos2A + 1 – sin2B – cos A cos B

= 1 + (cos2A – sin2B) – cos A cos B

= 1 + cos (A + B) · cos (A – B) – cos A cos B  ...[∵ cos (A + B) · cos (A – B) = cos2A – sin2B]

= `1 + cos  pi/3cos("A" - "B") - cos"A" cos"B"` ...[By (1)]

= `1 + 1/2cos("A" - "B") - cos"A" cos"B"`

= `1 + 1/2(cos"A" cos"B" + sin"A" sin"B") - cos"A" cos"B"`

= `1 + 1/2cos"A"cos"B" + 1/2sin "A" sin"B" - cos"A"cos"B"`

= `1 + 1/2sin"A"sin"B" - 1/2cos"A"cos"B"`

= `1 - 1/2(cos"A" cos"B" - sin"A"sin"B")`

= `1 - 1/2cos("A" + "B")`

= `1 - 1/2cos  pi/3`  ...[By (1)]

= `1 - 1/2 xx 1/2`

= `1 - 1/4`

= `3/4`

= R.H.S.

Concept: Factorization Formulae - Trigonometric Functions of Angles of a Triangle
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APPEARS IN

Balbharati Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 3 Trigonometry - 2
Miscellaneous Exercise 3 | Q II. (31) | Page 58
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