Question

Find the value of ${{\sin }^{2}}30{}^\circ +{{\cos }^{2}}30{}^\circ +{{\cot }^{2}}45{}^\circ $.

Answer 1

Hint: Try to simplify the expression by using the formula ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ , followed by putting the value of $\cot 45{}^\circ $ , that is known to us.

Complete step-by-step solution -
Before moving to the solution, let us discuss the periodicity of sine and cosine function, which we would be using in the solution. All the trigonometric ratios, including sine and cosine, are periodic functions. We can better understand this using the graph of sine and cosine.
First, let us start with the graph of sinx.

Next, let us see the graph of cosx.

Looking at both the graphs, we can say that the graphs are repeating after a fixed period i.e. $2{{\pi }^{c}}$ . So, we can say that the fundamental period of the cosine function and the sine function is $2{{\pi }^{c}}=360{}^\circ $
We will now solve the expression given in the question.
${{\sin }^{2}}30{}^\circ +{{\cos }^{2}}30{}^\circ +{{\cot }^{2}}45{}^\circ $
Now we know ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ . On using this formula in our expression, we get
$1+{{\cot }^{2}}45{}^\circ $
Now, as $45{}^\circ $ is a standard angle, we know that the value of $\cot 45{}^\circ =1$ . Putting this in our expression, we get
1+1
$=2$
Therefore, the value of ${{\sin }^{2}}30{}^\circ +{{\cos }^{2}}30{}^\circ +{{\cot }^{2}}45{}^\circ $ is 2.

Note: You need to remember the properties related to complementary angles and trigonometric ratios. It is preferred that while dealing with questions as above, you must first try to observe the pattern of the consecutive terms before applying the formulas, as directly applying the formulas may complicate the question.