Question

The value of the expression \[{{\sec }^{2}}\theta +\text{cose}{{\text{c}}^{2}}\theta \] is equal to
(A) \[{{\sec }^{2}}\theta .{{\cot }^{2}}\theta \]
(B) \[{{\sec }^{2}}\theta .{{\tan }^{2}}\theta \]
(C) \[\text{cose}{{\text{c}}^{2}}\theta .{{\cot }^{2}}\theta \]
(D) \[{{\sec }^{2}}\theta .\text{cose}{{\text{c}}^{2}}\theta \]

Answer 1

Hint: First of all, modify the given expression into sine and cosine terms using the identities \[\sec \theta =\dfrac{1}{\cos \theta }\] and \[\text{cosec}\theta \text{=}\dfrac{1}{\sin \theta }\] . Now, simplify it further and use the identity \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\] . At last, use the same identities \[\sec \theta =\dfrac{1}{\cos \theta }\] and \[\text{cosec}\theta \text{=}\dfrac{1}{\sin \theta }\] to get the result in terms of \[\sec \theta \] and \[\text{cosec}\theta \] .

Complete step-by-step solution:
According to the question, we are given a trigonometric expression and we have to calculate its value.
The given expression = \[{{\sec }^{2}}\theta +\text{cose}{{\text{c}}^{2}}\theta \] ……………………………..(1)
We can observe that the above equation needs to be simplified into a simpler form.
We know the identity that the secant of an angle is the reciprocal of the cosine of that angle i.e., \[\sec \theta =\dfrac{1}{\cos \theta }\] ………………………………….(2)
We also know the identity that the cosecant of an angle is the reciprocal of the sine of that angle i.e., \[\text{cosec}\theta \text{=}\dfrac{1}{\sin \theta }\] …………………………………..(3)
 Now, using equation (2) and equation (3), and on simplifying equation (1), we get
\[\begin{align}
  & ={{\sec }^{2}}\theta +\text{cose}{{\text{c}}^{2}}\theta \\
 & =\dfrac{1}{{{\cos }^{2}}\theta }+\dfrac{1}{{{\sin }^{2}}\theta } \\
\end{align}\]
\[=\dfrac{{{\sin }^{2}}\theta +{{\cos }^{2}}\theta }{{{\cos }^{2}}\theta {{\sin }^{2}}\theta }\] …………………………………….(4)
In the numerator of the above equation, we have the term \[\left( {{\sin }^{2}}\theta +{{\cos }^{2}}\theta \right)\] which can be simplified easily.
We also know the identity, \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\] …………………………………..(5)
Now, from equation (4) and equation (5), we get
\[=\dfrac{1}{{{\cos }^{2}}\theta {{\sin }^{2}}\theta }\] …………………………………………..(6)
In the above equation, we have sine and cosine terms but we don’t have any option having sine and cosine terms. So, we have to modify it into some other trigonometric terms.
Using equation (2) and equation (3), and on simplifying equation (6), we get
\[={{\sec }^{2}}\theta .\text{cose}{{\text{c}}^{2}}\theta \] ……………………………………..(7)
Therefore, the value of the given expression \[{{\sec }^{2}}\theta +\text{cose}{{\text{c}}^{2}}\theta \] is \[{{\sec }^{2}}\theta .\text{cose}{{\text{c}}^{2}}\theta \].
Hence, the correct option is (D).

Note: The best way to approach this type of question where we are provided an expression having two or three trigonometric terms and we are asked its value, is to convert all the terms into sine and cosine terms using proper identities. After conversion into sine and cosine terms, our calculation becomes easy.