Prove $\sec {x^2} + \cos ec{x^2} = \sec {x^2}.\cos ec{x^2}$

Hint: The above problem is based on the Properties of the trigonometric functions.
secx is the reciprocal of cosine and cosec x is the reciprocal of sine function of trigonometry
Using the concept of reciprocal property of sine and cosine functions, we will prove the given trigonometric function.

Complete step-by-step solution:
Let's discuss the sine and cosine functions first and then we will do the calculation.
Sine function of trigonometry is equal to perpendicular upon hypotenuse of the right triangle having some angle theta, which is an acute angle(angle less than ninety degree).
Cosine function is the trigonometric function which is equal to the base of the right triangle upon hypotenuse having an acute angle made with the base of the triangle.
Now, we will come to the calculation part of the problem.
We will solve the LHS first which we will bring equal to RHS.
$ \Rightarrow \sec {x^2} + \cos ec{x^2}$
We will convert the above written equation into sine and cosine functions;
$ \Rightarrow \dfrac{1}{{\cos {x^2}}} + \dfrac{1}{{\sin {x^2}}}$
LCM is taken and then we will use the property ($sinx^2 + cosx^2$ =$1$)
$ \Rightarrow \dfrac{{\cos {x^2} + \sin {x^2}}}{{\cos {x^2}.\sin {x^2}}}$ (when property which is written above is used, then numerator will become 1)
$ \Rightarrow \dfrac{1}{{\sin {x^2}.\cos {x^2}}}$ (again we will convert sine and cosine function into cosec and sec function)
$ \Rightarrow \sec {x^2}.\cos ec{x^2}$

Thus, RHS and LHS are equal and hence proved.

Note: Sine function is commonly used to model periodic phenomena such as sound and light waves, the position and the velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. Similarly, cosine function is used as a power factor function of the real power consumed by the system or on the consumer end.