Question

Find the value of \[(1 + \tan \theta + \sec \theta )(1 + \cot \theta - \operatorname{cosec} \theta )\].
(a) 0
(b) 1
(c) 2
(d) – 1

Answer 1

Hint: Recall the relations between the different trigonometric functions such as tanx, secx, cosecx, and cotx. Use the formula \[\cot x\tan x = 1\] to simplify cotangent and tangent terms. Convert every term into sine and cosine to simplify the expression.

Complete step-by-step answer:
A function of an angle expressed as the ratio of two of the sides of a right triangle that contains that angle is called trigonometric functions.
The sine, cosine, tangent, cotangent, secant, and cosecant are the trigonometric functions.
The sine and cosecant are inverses of each other. The cosine and secant are inverses of each other. The tangent and the cotangent are inverses of each other.
We are given the expression as follows:
\[T = (1 + \tan \theta + \sec \theta )(1 + \cot \theta - \operatorname{cosec} \theta )\]
We multiple the terms in the first bracket with the terms in the second bracket and we get three terms respectively as follows:
\[{{\text{T}}_1}{\text{ = }}1 + \cot \theta - {\text{cosec}}\theta .............(1)\]
\[{{\text{T}}_2}{\text{ = }}\tan \theta {\text{(}}1 + \cot \theta - {\text{cosec}}\theta )\]
\[{{\text{T}}_3}{\text{ = }}\sec \theta {\text{(}}1 + \cot \theta - {\text{cosec}}\theta )\]
Evaluating the second term, we have:
\[{{\text{T}}_2}{\text{ = tan}}\theta + {\text{tan}}\theta \cot \theta - {\text{tan}}\theta {\text{cosec}}\theta \]
We know that \[{\text{tan}}\theta \cot \theta = 1\] and \[{\text{tan}}\theta {\text{cosec}}\theta = \sec \], then we have as follows:
\[{{\text{T}}_2}{\text{ = }}\tan \theta + 1 - {\text{sec}}\theta .............(2)\]
We now simplify the third term \[{{\text{T}}_3}{\text{ = }}\sec \theta {\text{(}}1 + \cot \theta - {\text{cosec}}\theta )\].
\[{{\text{T}}_3}{\text{ = }}\sec \theta + \sec \theta \cot \theta - \sec \theta {\text{cosec}}\theta \]
We know that, \[\sec \theta \cot \theta = {\text{cosec}}\theta \], then we have as follows:
\[{{\text{T}}_3}{\text{ = }}\sec \theta + {\text{cosec}}\theta - \sec \theta {\text{cosec}}\theta ...........(3)\]
Adding equation (1), equation (2), and equation (3), we have:
\[T = 1 + \cot \theta - {\text{cosec}}\theta + \tan \theta + 1 - {\text{sec}}\theta + \sec \theta + {\text{cosec}}\theta - \sec \theta {\text{cosec}}\theta \]
Simplifying, we get:
\[T = 2 + \cot \theta + \tan \theta - \sec \theta {\text{cosec}}\theta \]
We write the above equation, in terms of sine and cosine and solve it.
\[T = 2 + \dfrac{{\cos \theta }}{{\sin \theta }} + \dfrac{{\sin \theta }}{{\cos \theta }} - \dfrac{1}{{\sin \theta {\text{cos}}\theta }}\]
Taking common denominator and solving, we get:
\[T = 2 + \dfrac{{{{\cos }^2}\theta + {{\sin }^2}\theta - 1}}{{\sin \theta \cos \theta }}\]
We know that \[{\cos ^2}\theta + {\sin ^2}\theta = 1\], then, we have:
\[T = 2 + \dfrac{{1 - 1}}{{\sin \theta \cos \theta }}\]
\[T = 2 + 0\]
\[T = 2\]
Hence, we have as follows:
\[(1 + \tan \theta + \sec \theta )(1 + \cot \theta - \operatorname{cosec} \theta ) = 2\]
Hence, the option (c) is the correct answer.

Note: If you make a mistake during substitution, you might end up with the answer as zero, which is wrong. You can also convert all terms into sine and cosine at the beginning itself and solve or you might solve the entire question without using them.