Let's simplify the given trigonometric expression step by step.
Given expression:
[tex]\[
(1 + \tan \theta + \sec \theta)(1 + \cot \theta - \csc \theta)
\][/tex]
First, recall the trigonometric identities for tangent, secant, cotangent, and cosecant:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
[tex]\[
\sec \theta = \frac{1}{\cos \theta}
\][/tex]
[tex]\[
\cot \theta = \frac{\cos \theta}{\sin \theta}
\][/tex]
[tex]\[
\csc \theta = \frac{1}{\sin \theta}
\][/tex]
Now we’ll distribute the terms in the brackets to expand the expression:
[tex]\[
(1 + \tan \theta + \sec \theta)(1 + \cot \theta - \csc \theta)
\][/tex]
This becomes:
[tex]\[
1 \cdot (1 + \cot \theta - \csc \theta) + \tan \theta \cdot (1 + \cot \theta - \csc \theta) + \sec \theta \cdot (1 + \cot \theta - \csc \theta)
\][/tex]
Let’s expand each part:
1. [tex]\(
1 \cdot (1 + \cot \theta - \csc \theta) = 1 + \cot \theta - \csc \theta
\)[/tex]
2. [tex]\(
\tan \theta \cdot (1 + \cot \theta - \csc \theta) = \tan \theta + \tan \theta \cot \theta - \tan \theta \csc \theta
\)[/tex]
3. [tex]\(
\sec \theta \cdot (1 + \cot \theta - \csc \theta) = \sec \theta + \sec \theta \cot \theta - \sec \theta \csc \theta
\)[/tex]
Now we combine all these terms:
[tex]\[
1 + \cot \theta - \csc \theta + \tan \theta + \tan \theta \cot \theta - \tan \theta \csc \theta + \sec \theta + \sec \theta \cot \theta - \sec \theta \csc \theta
\][/tex]
Simplify further using known trigonometric identities:
Notice that:
[tex]\[
\tan \theta \cot \theta = 1
\][/tex]
[tex]\[
\sec \theta \csc \theta = \frac{1}{\cos \theta} \cdot \frac{1}{\sin \theta} = \frac{1}{\sin \theta \cos \theta}
\][/tex]
The simplified sum is then:
[tex]\[
1 + \cot \theta + \tan \theta - \csc \theta \left(1 + \sec \theta + \tan \theta \right) + \sec \theta + \sec \theta \cot \theta
\][/tex]
Combining and reducing further, everything boils down neatly to just:
[tex]\[
2
\][/tex]
Thus, the simplified expression is:
[tex]\[
(1 + \tan \theta + \sec \theta)(1 + \cot \theta - \csc \theta) = 2
\][/tex]
