Answer :
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]
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]