Question
You have a triangle RST. Prove SinR^2 + CosR^2 = 1 given ST is perpendicular to RT.
Answer
Firstly,
sin R = opposite/hypotenuse
and
cos R = adjacent/hypotenuse
Square and add, to obtain,
sin^2 R + cos^2 R = (ST)^2/(SR)^2 + (RT)^2/(SR)^2
sin^2 R + cos^2 R = [(ST)^2 + (RT)^2 ] / [(SR)^2] …(1)
Now, By Pythagoras’ theorem, we have, ( note that SR is the hypotenuse because right angle is at T ),
(SR)^2 = (ST)^2 + (RT)^2 …(2)
Substitute (2) into (1), to get,
sin^2 R + cos^2 R = (SR)^2 / (SR)^2
sin^2 R + cos^2 R = 1
QED
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