Question

If A and B are complementary angles then ${\sin ^2}A + {\sin ^2}B =$
A. $0$
B. $\dfrac{1}{2}$
C. $2$
D.None of these

Answer 1

Hint : Always remember that the sum of two complementary angles is always ninety degrees. Here we will use the properties of the complementary angle and then will place it in the given expression and will simplify for the required solution.

Complete step-by-step answer :
Given that A and B are two complementary angles.
$A + B = 90^\circ$
It can be re-written as –
$\Rightarrow A = 90^\circ - B$ ... (a)
Now, take the given expression –
${\sin ^2}A + {\sin ^2}B$
Place the values in the above equation from (a)
$\Rightarrow {\sin ^2}(90^\circ - B) + {\sin ^2}B$
From the identity: ${\sin ^2}30^\circ = {\cos ^2}60^\circ$
So, $\Rightarrow {\cos ^2}B + {\sin ^2}B$
Also, we know that : ${\cos ^2}\theta + {\sin ^2}\theta = 1$
$\Rightarrow {\cos ^2}B + {\sin ^2}B = 1$
Hence, from the given multiple choices – the correct option is D.
So, the correct answer is “Option D”.

Note : Always remember the difference between the complementary and supplementary angles. The sum of two angles in supplementary is always one-eighty degree. The most important property of sines and cosines is that their values lie between minus one and plus one. Every point on the circle is unit circle from the origin. So, the coordinates of any point are within one of zero as well.