If $\cos ec_{}^2\theta + \cot _{}^2\theta = 7$, what is the value of (in degrees) of $\theta $?
$\left( {\text{1}} \right){\text{ 15}}$
$\left( 2 \right){\text{ 30}}$
$\left( 3 \right){\text{ 45}}$
$\left( 4 \right){\text{ 60}}$

Hint: This question can be easily solved by using the formula of from which we get, $\cos ec_{}^2\theta = 1 + \cot _{}^2\theta $ by moving $\cot _{}^2\theta $ to the right side. After getting the value of $\cos ec_{}^2\theta $ we can substitute it in the given equation $\cos ec_{}^2\theta + \cot _{}^2\theta = 7$ and by doing this we will get the value of $\theta $.

Formula used: $\cos ec_{}^2\theta - \cot _{}^2\theta = 1$
$\cot 30_{}^0 = \sqrt 3 $

Complete step-by-step answer:
It is given that $\cos ec_{}^2\theta + \cot _{}^2\theta = 7....\left( 1 \right)$
Since we know that $\cos ec_{}^2\theta - \cot _{}^2\theta = 1$
From this formula, we can write the value of $\cos ec_{}^2\theta $ by moving $\cot _{}^2\theta $ to the right hand side
 $\cos ec_{}^2\theta = 1 + \cot _{}^2\theta $
Now by putting the value of $\cos ec_{}^2\theta = 1 + \cot _{}^2\theta $ in equation $\left( 1 \right)$
$1 + \cot _{}^2\theta $ + $\cot _{}^2\theta = 7$
On adding the terms we get-
$1 + 2\cot _{}^2\theta = 7$
Now by moving the term $1$ on the right hand side we get-
$2\cot _{}^2\theta = 7 - 1$
After doing subtraction we get-
$2\cot _{}^2\theta = 6$
Now for getting the value of $\cot _{}^2\theta $ so we have to divide $6$ by $2$ and we get
$\cot _{}^2\theta = 3$
In order to get the value of $\cot \theta $, taking square on both side
$\cot \theta = \sqrt 3 $
Since the value of $\cot 30_{}^0 = \sqrt 3 $ so we can write
$\cot \theta = \cot 30_{}^0$
Therefore by applying cancellation technique we get
$\theta = 30_{}^0$
Thus the value of $\cos ec_{}^2\theta + \cot _{}^2\theta = 7$ is $30_{}^0$

So, the correct answer is “Option 2”.

Note: There are two ways of solving this question, one is by applying a substitution method and other is elimination method.
In both the cases we have to apply the formula of $\cos ec_{}^2\theta - \cot _{}^2\theta = 1$.
In case of substitution method you need substitute the value of $\cos ec_{}^2\theta $ in the given equation and can find out the value of $\theta $
But in the elimination method you need to eliminate one and find the value of the other and at last after getting the value of one you will be able to get the value of $\theta $.