Short Note

In a right angled triangle *ABC*, write the value of sin^{2} *A* + Sin^{2} *B* + Sin^{2} *C*.

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#### Solution

\[Let, \angle B = 90°\]

\[ \therefore A + C = 90°= \frac{\pi}{2}\]

\[ \Rightarrow C = \frac{\pi}{2} - A\]

\[ \Rightarrow \sin C = \sin \left( \frac{\pi}{2} - A \right)\]

\[ \Rightarrow \sin C = \cos A . . . \left( i \right)\]

\[\text{ Now,} \]

\[ \sin^2 A + \sin^2 B + \sin^2 C = \sin^2 A + 1 + \sin^2 C \left( \because \sin B = \sin 90°= 1 \right)\]

\[ = \sin^2 A + \cos^2 A + 1 \left[ \text{ Using } \left( i \right) \right]\]

\[ = 1 + 1\]

\[ = 2\]

Concept: Values of Trigonometric Functions at Multiples and Submultiples of an Angle

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