In a right angled tringle ABC, write the value of ^2A+ ^2B+ ^2C. - Vidyadip

Question

In a right angled tringle ABC, write the value of $\sin^2\text{A}+\sin^2\text{B}+\sin^2\text{C}.$

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Answer

Suppose in $\text{ABC}\angle\text{B}=90^\circ$
$\Rightarrow\text{A+C}=\frac{\pi}{2}$
$\Rightarrow\text{A}=\frac{\pi}{2}-\text{C}$
$\Rightarrow\sin\text{A}=\sin\Big(\frac{\pi}{2}-\text{C}\Big)$
Now,
$\sin^2\text{A}+\sin^2\text{B}+\sin^2\text{C}$
$=\sin^2\text{A}+1+\cos^2\text{A}$ $\big[\because\sin\frac{\pi}{2}=1\big]$
$=1+1=2$

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