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Question: Find the value of \({{\sec }^{2}}\left( \dfrac{\pi }{4} \right)\)?...

Find the value of sec⁡2(π4){{\sec }^{2}}\left( \dfrac{\pi }{4} \right)?

Explanation

Solution

Here we have to find the value of trigonometric function at an angle given. Firstly we will convert the given secant function into cosine function by using the inverse relation between them. Then we will substitute the value of the cosine function at that angle which is commonly known to us. Finally we will simplify the value obtained to get our desired answer.

Complete answer:
We have to find the value of the function given as follows:
sec⁡2(π4){{\sec }^{2}}\left( \dfrac{\pi }{4} \right)….(1)\left( 1 \right)
As we know that secant is inverse function of cosine and their relation is given as follows:
sec⁡x=1cos⁡x\sec x=\dfrac{1}{\cos x}
On squaring both sides we get,
sec⁡2x=1cos⁡2x{{\sec }^{2}}x=\dfrac{1}{{{\cos }^{2}}x}
Using the above relation in equation (1) we get,
⇒sec⁡2(π4)=(1cos⁡2(π4))\Rightarrow {{\sec }^{2}}\left( \dfrac{\pi }{4} \right)=\left( \dfrac{1}{{{\cos }^{2}}\left( \dfrac{\pi }{4} \right)} \right)
⇒sec⁡2(π4)=(1cos⁡(π4))2\Rightarrow {{\sec }^{2}}\left( \dfrac{\pi }{4} \right)={{\left( \dfrac{1}{\cos \left( \dfrac{\pi }{4} \right)} \right)}^{2}}….(2)\left( 2 \right)
We know that value of cosine at π4\dfrac{\pi }{4}is given as:
cos⁡(π4)=12\cos \left( \dfrac{\pi }{4} \right)=\dfrac{1}{\sqrt{2}}
So using it equation (1) we get
⇒sec⁡2(π4)=(112)2\Rightarrow {{\sec }^{2}}\left( \dfrac{\pi }{4} \right)={{\left( \dfrac{1}{\dfrac{1}{\sqrt{2}}} \right)}^{2}}
⇒sec⁡2(π4)=112\Rightarrow {{\sec }^{2}}\left( \dfrac{\pi }{4} \right)=\dfrac{1}{\dfrac{1}{2}}
So we get the value as:
⇒sec⁡2(π4)=2\Rightarrow {{\sec }^{2}}\left( \dfrac{\pi }{4} \right)=2
Hence the value of sec⁡2(π4){{\sec }^{2}}\left( \dfrac{\pi }{4} \right) is 22 .

Note:
Trigonometry is a very important branch of mathematics which deals with the angles and sides of a right-angled triangle. The basic trigonometry functions are as sine, cosine, tangent, cosecant, secant and cotangent. The last three functions are the inverse of the first three functions respectively and we have used this inverse relation for solving our question. The sign π=180∘\pi ={{180}^{\circ }} in angles and the values of trigonometric function at some angles are commonly known which we have used in this question to get our answer. The angles are usually measured in degrees and radians. There are many different relations and identities among the six trigonometric functions which are used when the question is more complex.