Question

The value of ${{\sin }^{2}}25{}^\circ +{{\sin }^{2}}65{}^\circ $ is equal to
A. $0$
B. $2{{\sin }^{2}}25{}^\circ $
C. $2{{\cos }^{2}}65{}^\circ $
D. $1$

Answer 1

Hint: In this problem we will use the trigonometric identity ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$. In this problem we will convert the one trigonometric ratio into another trigonometric ratio by writing one angle into its complementary/supplementary and use the All Silver Tea Cups rule. Now we will convert the obtained equation into trigonometric identity.

Complete step by step answer:
Given that, ${{\sin }^{2}}25{}^\circ +{{\sin }^{2}}65{}^\circ $.
Let $x=\sin 25{}^\circ $, $y=\sin 65{}^\circ $ then ${{\sin }^{2}}25{}^\circ +{{\sin }^{2}}65{}^\circ ={{x}^{2}}+{{y}^{2}}$
We have the relation between the given angles $25{}^\circ ,65{}^\circ $ as $25{}^\circ +65{}^\circ =90{}^\circ $ i.e. the given two angles are the complementary angles so we can write $25{}^\circ =90{}^\circ -65{}^\circ $ or $65{}^\circ =90{}^\circ -25{}^\circ $
Now the value of $x$ is
$x=\sin \left( 90{}^\circ -65{}^\circ \right)$
From the rule of All Silver Tea Cups we have $\sin \left( 90{}^\circ -\theta \right)=\cos \theta $, then we will get
$x=\cos 65{}^\circ $
Now the value of ${{x}^{2}}+{{y}^{2}}$ is given by
${{x}^{2}}+{{y}^{2}}={{\cos }^{2}}65{}^\circ +{{\sin }^{2}}65{}^\circ $
We have the trigonometric identity ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$, then the value of ${{x}^{2}}+{{y}^{2}}$ is
$\begin{align}
  & {{x}^{2}}+{{y}^{2}}=1 \\
 & \Rightarrow {{\sin }^{2}}25{}^\circ +{{\sin }^{2}}65{}^\circ =1 \\
\end{align}$

So, the correct answer is “Option D”.

Note: We can also use $65{}^\circ =90{}^\circ -25{}^\circ $ in $y$, then the value of $y$ is given by
$y=\sin \left( 90{}^\circ -25{}^\circ \right)$
From the rule of All Silver Tea Cups we have $\sin \left( 90{}^\circ -\theta \right)=\cos \theta $, then we will get
$y=\cos 25{}^\circ $
Now the value of ${{x}^{2}}+{{y}^{2}}$ is given by
${{x}^{2}}+{{y}^{2}}={{\sin }^{2}}25{}^\circ +{{\cos }^{2}}25{}^\circ $
We have the trigonometric identity ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$, then the value of ${{x}^{2}}+{{y}^{2}}$ is
$\begin{align}
  & {{x}^{2}}+{{y}^{2}}=1 \\
 & \Rightarrow {{\sin }^{2}}25{}^\circ +{{\sin }^{2}}65{}^\circ =1 \\
\end{align}$
$\therefore $ Option – D is the correct answer. From both the methods we got the same result.
Some other trigonometric identities that are useful in solving this type of problems are listed below
${{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1$
${{\csc }^{2}}\theta -{{\cot }^{2}}\theta =1$.