First slide

Trigonometric transformations

Question

If $\cos^{2} A + \cos^{2} B + \cos^{2} C = 1$ , then $∆ ABC$ is

Moderate

A

equilateral
B

isosceles
C

right angled
D

none of these

Solution

We have $\cos^{2} A + \cos^{2} B - (1 - \cos^{2} C) = 0$
$\begin{array}{ll} or \cos^{2} A + \cos^{2} B - \sin^{2} C = 0 \\ or \cos^{2} A + \cos (B + C) \cos (B - C) = 0 \\ or \cos A [\cos A - \cos (B - C)] = 0 \\ or \cos A [\cos (B + C) + \cos (B - C)] = 0 \\ or 2 \cos Acos Bcos C = 0 \end{array}$
Hence, either $A or B or C is 90^{\circ} .$

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