(C) Expand the expression: $(a^2 + b^2 - 2ab) \cos^2 \frac{C}{2} + (a^2 + b^2 + 2ab) \sin^2 \frac{C}{2}$ $= (a^2 + b^2)(\cos^2 \frac{C}{2} + \sin^2 \f

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(C) Expand the expression: $(a^2 + b^2 - 2ab) \cos^2 \frac{C}{2} + (a^2 + b^2 + 2ab) \sin^2 \frac{C}{2}$ $= (a^2 + b^2)(\cos^2 \frac{C}{2} + \sin^2 \frac{C}{2}) + 2ab(\sin^2 \frac{C}{2} - \cos^2 \frac{C}{2})$ $= (a^2 + b^2)(1) - 2ab(\cos^2 \frac{C}{2} - \sin^2 \frac{C}{2})$ $= a^2 + b^2 - 2ab \cos C$ Using the Law of Cosines,$c^2 = a^2 + b^2 - 2ab \cos C$,so the expression equals $c^2$.

Class 11 Mathematics Trigonometrical Equations Mix Examples-Trigonometrical Equations and Inequations, Properties of Triangles, Height and Distance

Easy

In $\Delta ABC$,$(a - b)^2 \cos^2 \frac{C}{2} + (a + b)^2 \sin^2 \frac{C}{2} = $

A
$a^2$
B
$b^2$
C
$c^2$
D
None of these

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Trigonometrical Equations

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Mix Examples-Trigonometrical Equations and Inequations, Properties of Triangles, Height and Distance

If $A$ is the area and $2s$ is the sum of $3$ sides of a triangle,then:

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In $\triangle ABC$,if $a, b, c$ are $5, 12$ and $13$ respectively,then $b^2 \sin 2C + c^2 \sin 2B =$

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If $A, B, C$ are the angles of a triangle,then $\sin^2 A + \sin^2 B + \sin^2 C - 2\cos A \cos B \cos C = $

Medium

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The sum of the sides of a right-angled triangle is $42$,and the difference between the median and altitude drawn from the vertex at the right angle is $2$. The area of the triangle is

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In $\Delta ABC$,$(a - b)^2 \cos^2 \frac{C}{2} + (a + b)^2 \sin^2 \frac{C}{2} = $

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