Question

Find the value of ${\sec ^2}({\tan ^{ - 1}}2) + \cos e{c^2}({\cot ^{ - 1}}3) =$
A) 5
B) 13
C)15
D) 6

Answer 1

To solve this question, you have to know about trigonometric equations. First of all we will use the basic formula of ${\sec ^2}\theta$ and $\cos e{c^2}\theta$ . We use this formula to simplify our equation in an easy way. Then we will put one property of $\tan \theta$ and $\cot \theta$ for further simplification and after that we just calculate the basic math to get our final answer.

Complete step by step answer:
First of all let’s see our given equation,
$\Rightarrow {\sec ^2}({\tan ^{ - 1}}2) + \cos e{c^2}({\cot ^{ - 1}}3)$
So, our given equation is in terms of ${\sec ^2}\theta$ and $\cos e{c^2}\theta$ , so we have to use basic trigonometric equation to convert it into easy equation,
We use,
$\Rightarrow {\sec ^2}\theta = 1 + {\tan ^2}\theta$ and
$\Rightarrow \cos e{c^2}\theta = 1 + {\cot ^2}\theta$
So, now put above values of ${\sec ^2}\theta$ and $\cos e{c^2}\theta$ in our given equation and we will get,
$\Rightarrow 1 + {\tan ^2}({\tan ^{ - 1}}2) + 1 + {\cot ^2}({\cot ^{ - 1}}3)$
We knew that $\tan^2 x = (\tan x)^2$, $\cot^2 x = (\cot x)^2$ and From inverse trigonometry we knew that $\tan (\tan ^{-1} x) = x$, $\cot (\cot^{-1} x=x$
This implies that ${\tan ^2}({\tan ^{ - 1}}x) = x^2$ and ${\cot ^2}({\cot ^{ - 1}}x) = x^2$ so let’s use this formula in above equation and we will get,
$\Rightarrow 1 + {(2)^2} + 1 + {(3)^2}$
Now, just do simple mathematics to get our final answer,
$\Rightarrow 1 + 4 + 1 + 9$
Adding these values we get
$\Rightarrow 15$
Therefore, the value of ${\sec ^2}({\tan ^{ - 1}}2) + \cos e{c^2}({\cot ^{ - 1}}3) = 15$. So, option (C) is correct.

Note:
We can do this question in another way also but in that way we have to remember values of a given angle. In this question values are simple i.e. 2 and 3. So we can use trigonometric triangles to find the value of it, but it will be hard when the given angle value is not as common as we saw in this problem. So it’s better to approach this question with the above method so you don’t need to remember any values.