Question

In ΔABC, `(a - b)^2 cos^2  C/2 + (a + b)^2 sin^2  C/2` is equal to ______.

विकल्प

• b²

• c²

• a²

• a² + b² + c²

Answer 1

In ΔABC, `(a - b)^2 cos^2  C/2 + (a + b)^2 sin^2  C/2` is equal to c².

Explanation:

`(a - b)^2 cos^2  C/2 + (a + b)^2 sin^2  C/2`

= `(a^2 + b^2 - 2ab) cos^2  C/2 + (a^2 + b^2 + 2ab) sin^2  C/2`

= `a^2(cos^2  C/2 + sin^2  C/2) + b^2(cos^2  C/2 + sin^2  C/2) - 2ab(cos^2  C/2 - sin^2  C/2)`

= a² + b² – 2ab cos C  ...`[∵ cos^2  C/2 - sin^2  C/2 = cos C]`

= `a^2 + b^2 - 2ab((a^2 + b^2 - c^2)/(2ab))` = c²  ...`[∵ cos C = (a^2 + b^2 - c^2)/(2ab)]`