First slide

Trigonometric transformations

Question

If $A + B + C = π / 2$ , then $\sin^{2} A + \sin^{2} B + \sin^{2} C + 2 \sin A \sin B \sin C$ is equal to

Moderate

A

0
B

1
C

-1
D

none of these

Solution

$\begin{array}{l} \sin^{2} A + \sin^{2} B + \sin^{2} C + 2 \sin A \sin B \sin C \\ = 1 - (\cos^{2} A - \sin^{2} B) + \sin C [\sin C + 2 \sin A \sin B] \\ = 1 - \cos (A + B) \cos (A - B) + \sin C [\cos (A + B) + 2 \sin A \sin B] \\ [∵ C = π / 2 - A - B] \\ = 1 - \cos (π / 2 - C) \cos (A - B) + \sin C \cos (A - B) \\ = 1 \end{array}$

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