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

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.