Question

How do you evaluate ${\sec ^2}\left( {\dfrac{\pi }{4}} \right)$ ?

Answer 1

For solving this question we just need one formula and it is given by $\sec x = \dfrac{1}{{\cos x}}$ and as we know that the value of $\cos 45$ is equal to $\dfrac{{\sqrt 2 }}{2}$ . So by substituting these values and solving them we will be able to get the result.

Formula used:
The trigonometric formula used is
$\sec x = \dfrac{1}{{\cos x}}$

Complete step by step answer:
As we have the question given by $\sec 45$
Now for solving this firstly we will convert the above trigonometric function in the form of sine. So by using the formula we can write the trigonometric function as
$\Rightarrow \sec x = \dfrac{1}{{\cos x}}$
Now on substituting the value of $x$ , we will get the equation as
$\Rightarrow \sec 45 = \dfrac{1}{{\cos 45}}$
As we know that the value of $\cos 45$ is equal to $\dfrac{{\sqrt 2 }}{2}$ .
Therefore, on substituting the values, we will get the equation as
$\Rightarrow \sec 45 = \dfrac{1}{{\dfrac{{\sqrt 2 }}{2}}}$
And on solving it we will get
$\Rightarrow \sec 45 = \dfrac{2}{{\sqrt 2 }}$
By doing the multiplication and division by $\sqrt 2$ in the right side, we will get the equation as
$\Rightarrow \sec 45 = \dfrac{2}{{\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }}$
And on solving it we will get the equation as
$\Rightarrow \sec 45 = \dfrac{{2\sqrt 2 }}{2}$
Since the liker term will be canceled, so we will get the equation as
$\Rightarrow \sec 45 = \sqrt 2$
Now on squaring both the sides, we get the equation as
$\Rightarrow {\sec ^2}\left( {\dfrac{\pi }{4}} \right) = 2$

Hence, the value of $\sec 45$ will is equal to $2$.

Note: For solving a question like this or any type of question where we need to change the equation in terms of other trigonometric identities then we should always convert the equations either in cosine or sine function and then we can easily solve such types of questions.