Question

Prove the following trigonometric equation:
\[{{\sin }^{2}}\dfrac{\pi }{6}+{{\cos }^{2}}\dfrac{\pi }{3}-{{\tan }^{2}}\dfrac{\pi }{4}=-\dfrac{1}{2}\]

Answer 1

Hint: For the above question we will use the values of trigonometric function at the given angle which is shown as follows:
\[\begin{align}
  & \sin \dfrac{\pi }{6}=\dfrac{1}{2} \\
 & \cos \dfrac{\pi }{3}=\dfrac{1}{2} \\
 & \tan \dfrac{\pi }{4}=1 \\
\end{align}\]
Then, we can substitute these in the LHS of the given equation and then prove it.

Complete step-by-step answer:
We have been asked to prove the following equation: \[{{\sin }^{2}}\dfrac{\pi }{6}+{{\cos }^{2}}\dfrac{\pi }{3}-{{\tan }^{2}}\dfrac{\pi }{4}=-\dfrac{1}{2}\]
We can see that all the angles in the equation are standard angles whose values are known to us. Now we know that values of the above trigonometric ratios for the given angles are given by:
\[\sin \dfrac{\pi }{6}=\dfrac{1}{2},\cos \dfrac{\pi }{3}=\dfrac{1}{2},\tan \dfrac{\pi }{4}=1\]
Now considering the left hand side of the given expression, substituting the values and simplifying the equation, we get as follows:
\[{{\sin }^{2}}\dfrac{\pi }{6}+{{\cos }^{2}}\dfrac{\pi }{3}-{{\tan }^{2}}\dfrac{\pi }{4}={{\left( \dfrac{1}{2} \right)}^{2}}+{{\left( \dfrac{1}{2} \right)}^{2}}-{{\left( 1 \right)}^{2}}=\dfrac{1}{4}+\dfrac{1}{4}-1\]
On taking the LCM of the terms we get as follows:
\[{{\sin }^{2}}\dfrac{\pi }{6}+{{\cos }^{2}}\dfrac{\pi }{3}-{{\tan }^{2}}\dfrac{\pi }{4}=\dfrac{1+1-4}{4}=\dfrac{2-4}{4}=\dfrac{-2}{4}=\dfrac{-1}{2}\]
Now we can see that we have got the result as = right hand side.
Hence, the left hand side of the expression is equal to the right hand side.
Therefore, the given expression is proved.

Note: Sometimes by mistake we use the values of \[\cos \dfrac{\pi }{3}\] as \[\dfrac{\sqrt{3}}{2}\] which is wrong and thus we get the incorrect result. So be careful while substituting all the trigonometric values in the expression. Also, try to memorize all the values for different angles which will help you a lot in these types of questions otherwise you will always have to take the help of the trigonometric ratio table. Apart from this, any mistake while simplifying and while taking the LCM should be avoided as it will lead to waste of time in the exams.