Question

Find the value of k for the given expression \[\left( {1 - \cot 22^\circ } \right)\left( {1 - \cot 23^\circ } \right) = \sqrt k \]

Answer 1

Hint: The given question deals with the concept of trigonometry. In order to solve this question we will take use trigonometric identity related to cot theta i.e., \[\cot (A + B) = \dfrac{{\cot A \times \cot B - 1}}{{\cot B + \cot A}}\]. We will assume \[\cot A = \cot 22^\circ \]and \[\cot B = \cot 23^\circ \] put these values in the trigonometric identity and solve it until we reach a conclusion.

Complete step by step solution:
Given that, \[\left( {1 - \cot 22^\circ } \right)\left( {1 - \cot 23^\circ } \right) = \sqrt k \]
We know that \[22^\circ + 23^\circ = 45^\circ \]
We have the given expression in cot theta, therefore we use identity related to cot theta i.e., \[\cot (A + B) = \dfrac{{\cot A \times \cot B - 1}}{{\cot B + \cot A}} - - - - - (1)\].
Now, let us assume, \[\cot A = \cot 22^\circ \]and \[\cot B = \cot 23^\circ \]
Here, put these values into the trigonometric identity above (1)
Thus, we have,
\[\cot \left( {22^\circ + 23^\circ } \right) = \dfrac{{\cot 22^\circ \times \cot 23^\circ - 1}}{{\cot 23^\circ + \cot 22^\circ }}\]
Which is,
\[ \Rightarrow 1 = \dfrac{{\cot 22^\circ \times \cot 23^\circ - 1}}{{\cot 23^\circ + \cot 22^\circ }}\]

As we know \[\cot 45^\circ = 1\] from the trigonometric table of values.Simplifying the above expression we get,
\[ \Rightarrow \cot 23^\circ + \cot 22^\circ = \cot 22^\circ \times \cot 23^\circ - 1\]
\[ \Rightarrow 1 = \cot 22^\circ \times \cot 23^\circ - \cot 23^\circ - \cot 22^\circ \]
Now, we add 1 to both the sides of the above expression
\[ \Rightarrow 1 + 1 = \cot 22^\circ \times \cot 23^\circ - \cot 23^\circ - \cot 22^\circ + 1\]
Rearranging the above expression further we get,
\[ \Rightarrow 2 = \cot 22^\circ \times \cot 23^\circ - \cot 22^\circ + 1 - \cot 23^\circ \]
Here, we take \[ - \cot 22^\circ \] common from the RHS of the above expression

We get,
\[ \Rightarrow 2 = - \cot 22^\circ \left( { - \cot 23^\circ + 1} \right) + \left( {1 - \cot 23^\circ } \right)\]
Further, taking the common factor from the above expression we get,
\[ \Rightarrow 2 = \left( { - \cot 23^\circ + 1} \right)\left( {1 - \cot 22^\circ } \right)\]
Rearranging the above expression
We get,
\[ \therefore 2 = \left( {1 - \cot 22^\circ } \right)\left( {1 - \cot 23^\circ } \right)\]
We know, \[\sqrt 4 = 2\] therefore, the value of k is 4.

Hence, the value of $k$ is $4$.

Note: The value of cot is listed in the standard trigonometric table of values. Trigonometric table consists of trigonometric ratios from 0 degrees to 360 degrees. These trigonometric ratios are:
Sine= Hypotenuse by base
Cosine= Base by hypotenuse
Tangent= Perpendicular by base
The other three ratios are cosecant, secant and cotangent and they are reciprocal to the above listed ratios respectively. The trigonometric table is as follows:

Angle in degrees	0	30	45	60	90	180	270	360
Sine	\[0\]	\[\dfrac{1}{2}\]	\[\dfrac{1}{{\sqrt 2 }}\]	\[\dfrac{{\sqrt 3 }}{2}\]	\[1\]	\[0\]	\[ - 1\]	\[0\]
Cosine	\[1\]	\[\dfrac{{\sqrt 3 }}{2}\]	\[\dfrac{1}{{\sqrt 2 }}\]	\[\dfrac{1}{2}\]	\[0\]	\[ - 1\]	\[0\]	\[1\]
Tangent	\[0\]	\[\dfrac{1}{{\sqrt 3 }}\]	\[1\]	\[\sqrt 3 \]	Not defined	\[0\]	Not defined	1
Cosecant	Not defined	\[2\]	\[\sqrt 2 \]	\[\dfrac{2}{{\sqrt 3 }}\]	\[1\]	Not defined	\[ - 1\]	Not defined
Secant	\[1\]	\[\dfrac{2}{{\sqrt 3 }}\]	\[\sqrt 2 \]	\[2\]	Not defined	\[ - 1\]	Not defined	\[1\]
cotangent	Not defined	\[\sqrt 3 \]	\[1\]	\[\dfrac{1}{{\sqrt 3 }}\]	\[0\]	Not defined	\[0\]	Not defined

Here, the value of cot theta we have used to solve the above question is derived from the above table.