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In Delta ABC, prove that (a - b)^(2) cos...

In `Delta ABC`, prove that `(a - b)^(2) cos^(2).(C)/(2) + (a + b)^(2) sin^(2).(C)/(2) = c^(2)`

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L.H.S. ` = ( a - b)^(2) cos^(2) ((C)/(2)) + (a + b)^(2) sin^(2) ((C)/(2))`
` = (a^(2) - 2ab + b^(2) ) cos^(2) ((C)/(2)) + (a^(2) + 2ab + b^(2)) sin^(2) ((C)/(2))`
` = a^(2) [ cos^(2) ((C)/(2)) + sin^(2) ((C)/(2)) ] - 2ab `
`[ cos^(2) ((C)/(2)) - sin^(2) ((C)/(2))]+ b^(2) [ cos^(2) ((C)/(2)) + sin^(2) ((C)/(2))]`
` = a^(2) . (1) - 2ab cos C + b^(2) `
` " " [because sin^(2) A + cos^(2) A = 1]`
` = a^(2) - 2ab cos C + b^(2)`
` = a^(2) + b^(2) - 2ab cos C `
` therefore L.H.S = c^(2) " "` ...(By cosine rule)
Hence Proved .
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