First slide

Trigonometric transformations

Question

In any triangle ABC, $\sin^{2} A - \sin^{2} B + \sin^{2} C$ is always equal to

Moderate

A

2 sinA sinB cos C
B

2 sinA cos.B sinC
C

2 sinA cosB cos C
D

2 sinA sinB sin C

Solution

$\begin{array}{l} \sin^{2} A - \sin^{2} B + \sin^{2} C \\ = \sin (A + B) \sin (A - B) + \sin^{2} C \\ = \sin C (\sin (A - B) + \sin C) \\ = \sin C (\sin (A - B) + \sin (A + B)) \\ = 2 \sin Acos Bsin C \end{array}$

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