Maharashtra State BoardHSC Arts 12th Board Exam
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In ∆ABC, prove that (b-c)2cos2(A2)+(b+c)2sin2(A2) = a2 - Mathematics and Statistics

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Sum

In ∆ABC, prove that `("b" - "c")^2 cos^2 ("A"/2) + ("b" + "c")^2 sin^2 ("A"/2)` = a2 

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Solution

L.H.S. = `("b" - "c")^2 cos^2  "A"/2 + ("b" + "c")^2 sin^2  "A"/2`

= `("b"^2 + "c"^2 - 2"bc") cos^2  "a"/2 + ("b"^2 + "c"^2 + 2"bc") sin^2  "A"/2`

= `("b"^2 + "c"^2) cos^2  "A"/2 - 2"bc" cos^2  "A"/2 + ("b"^2 + "c"^2) sin^2  "A"/2 + 2"bc" sin^2  "A"/2`

=`("b"^2  + "c"^2) (cos^2  "A"/2 + sin^2  "A"/2) - 2"bc" (cos^2  "A"/2 - sin^2  "A"/2)`

= (b2 + c2)(1) − 2bc cos A  .......[∵ cos2θ − sin2θ = cos 2θ]

= b2 + c2 − 2bc cos A

= a2    .......[By cosine rule]

= R.H.S.

Concept: Solutions of Triangle
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