Answer :
Let's simplify each given trigonometric expression step by step.
### Expression (a)
[tex]\[ \sec^2 \theta + \csc^2 \theta - \sec^2 \theta \cdot \csc^2 \theta \][/tex]
First, recall the definitions of the secant and cosecant functions:
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \quad \text{and} \quad \csc \theta = \frac{1}{\sin \theta} \][/tex]
Let's rewrite the expression using these definitions:
[tex]\[ \left(\frac{1}{\cos \theta}\right)^2 + \left(\frac{1}{\sin \theta}\right)^2 - \left(\frac{1}{\cos \theta}\right)^2 \cdot \left(\frac{1}{\sin \theta}\right)^2 \][/tex]
Simplify each term:
[tex]\[ \frac{1}{\cos^2 \theta} + \frac{1}{\sin^2 \theta} - \frac{1}{\cos^2 \theta \cdot \sin^2 \theta} \][/tex]
To combine these fractions, we need a common denominator:
[tex]\[ \frac{\sin^2 \theta + \cos^2 \theta - 1}{\cos^2 \theta \cdot \sin^2 \theta} \][/tex]
Since [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex], we have:
[tex]\[ \frac{1 - 1}{\cos^2 \theta \cdot \sin^2 \theta} = \frac{0}{\cos^2 \theta \cdot \sin^2 \theta} = 0 \][/tex]
Therefore, the expression simplifies to:
[tex]\[ 0 \][/tex]
### Expression (b)
[tex]\[ (\sec \theta - \cos \theta) (\csc \theta - \sin \theta) \][/tex]
Substitute the definitions:
[tex]\[ \left( \frac{1}{\cos \theta} - \cos \theta \right) \left( \frac{1}{\sin \theta} - \sin \theta \right) \][/tex]
Combine the fractions inside the parentheses:
[tex]\[ \left( \frac{1 - \cos^2 \theta}{\cos \theta} \right) \left( \frac{1 - \sin^2 \theta}{\sin \theta} \right) \][/tex]
Use the Pythagorean identity [tex]\(1 - \cos^2 \theta = \sin^2 \theta\)[/tex] and [tex]\(1 - \sin^2 \theta = \cos^2 \theta\)[/tex]:
[tex]\[ \left( \frac{\sin^2 \theta}{\cos \theta} \right) \left( \frac{\cos^2 \theta}{\sin \theta} \right) \][/tex]
Simplify the fraction:
[tex]\[ \sin \theta \cos \theta \][/tex]
Using the double-angle identity for sine, this simplifies to:
[tex]\[ \frac{\sin(2\theta)}{2} \][/tex]
### Expression (c)
[tex]\[ (1 - \cos^2 \theta) (1 + \cot^2 \theta) \][/tex]
Rewrite using the Pythagorean identities:
[tex]\[ \sin^2 \theta \left( 1 + \frac{\cos^2 \theta}{\sin^2 \theta} \right) \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ \sin^2 \theta \left( \frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta} \right) \][/tex]
Since [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex], this becomes:
[tex]\[ \sin^2 \theta \left( \frac{1}{\sin^2 \theta} \right) = 1 \][/tex]
### Expression (d)
[tex]\[ \frac{\sec^2 \theta - \csc^2 \theta}{2} \][/tex]
Rewrite using the definitions:
[tex]\[ \frac{\frac{1}{\cos^2 \theta} - \frac{1}{\sin^2 \theta}}{2} \][/tex]
Combine the fractions:
[tex]\[ \frac{\frac{\sin^2 \theta - \cos^2 \theta}{\cos^2 \theta \sin^2 \theta}}{2} \][/tex]
Simplify the numerator using the identity [tex]\(\sin^2 \theta - \cos^2 \theta = -(\cos^2 \theta - \sin^2 \theta)\)[/tex]:
[tex]\[ \frac{- (\cos^2 \theta - \sin^2 \theta)}{2 \cos^2 \theta \sin^2 \theta} \][/tex]
Factor out the negative sign:
[tex]\[ - \frac{\cos^2 \theta - \sin^2 \theta}{2 \cos^2 \theta \sin^2 \theta} \][/tex]
Split the fraction:
[tex]\[ - \frac{\cos^2 \theta}{2 \cos^2 \theta \sin^2 \theta} + \frac{\sin^2 \theta}{2 \cos^2 \theta \sin^2 \theta} \][/tex]
Simplify each term:
[tex]\[ - \frac{1}{2 \sin^2 \theta} + \frac{1}{2 \cos^2 \theta} \][/tex]
Rewrite each fraction:
[tex]\[ - \frac{\csc^2 \theta}{2} + \frac{\sec^2 \theta}{2} \][/tex]
So the expression simplifies to:
[tex]\[ - \frac{\csc^2 \theta}{2} + \frac{\sec^2 \theta}{2} \][/tex]
### Summary
a) [tex]\[ \sec^2 \theta + \csc^2 \theta - \sec^2 \theta \cdot \csc^2 \theta = 0 \][/tex]
b) [tex]\[ (\sec \theta - \cos \theta)(\csc \theta - \sin \theta) = \frac{\sin(2\theta)}{2} \][/tex]
c) [tex]\[ (1 - \cos^2 \theta)(1 + \cot^2 \theta) = 1 \][/tex]
d) [tex]\[ \frac{\sec^2 \theta - \csc^2 \theta}{2} = - \frac{\csc^2 \theta}{2} + \frac{\sec^2 \theta}{2} \][/tex]
### Expression (a)
[tex]\[ \sec^2 \theta + \csc^2 \theta - \sec^2 \theta \cdot \csc^2 \theta \][/tex]
First, recall the definitions of the secant and cosecant functions:
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \quad \text{and} \quad \csc \theta = \frac{1}{\sin \theta} \][/tex]
Let's rewrite the expression using these definitions:
[tex]\[ \left(\frac{1}{\cos \theta}\right)^2 + \left(\frac{1}{\sin \theta}\right)^2 - \left(\frac{1}{\cos \theta}\right)^2 \cdot \left(\frac{1}{\sin \theta}\right)^2 \][/tex]
Simplify each term:
[tex]\[ \frac{1}{\cos^2 \theta} + \frac{1}{\sin^2 \theta} - \frac{1}{\cos^2 \theta \cdot \sin^2 \theta} \][/tex]
To combine these fractions, we need a common denominator:
[tex]\[ \frac{\sin^2 \theta + \cos^2 \theta - 1}{\cos^2 \theta \cdot \sin^2 \theta} \][/tex]
Since [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex], we have:
[tex]\[ \frac{1 - 1}{\cos^2 \theta \cdot \sin^2 \theta} = \frac{0}{\cos^2 \theta \cdot \sin^2 \theta} = 0 \][/tex]
Therefore, the expression simplifies to:
[tex]\[ 0 \][/tex]
### Expression (b)
[tex]\[ (\sec \theta - \cos \theta) (\csc \theta - \sin \theta) \][/tex]
Substitute the definitions:
[tex]\[ \left( \frac{1}{\cos \theta} - \cos \theta \right) \left( \frac{1}{\sin \theta} - \sin \theta \right) \][/tex]
Combine the fractions inside the parentheses:
[tex]\[ \left( \frac{1 - \cos^2 \theta}{\cos \theta} \right) \left( \frac{1 - \sin^2 \theta}{\sin \theta} \right) \][/tex]
Use the Pythagorean identity [tex]\(1 - \cos^2 \theta = \sin^2 \theta\)[/tex] and [tex]\(1 - \sin^2 \theta = \cos^2 \theta\)[/tex]:
[tex]\[ \left( \frac{\sin^2 \theta}{\cos \theta} \right) \left( \frac{\cos^2 \theta}{\sin \theta} \right) \][/tex]
Simplify the fraction:
[tex]\[ \sin \theta \cos \theta \][/tex]
Using the double-angle identity for sine, this simplifies to:
[tex]\[ \frac{\sin(2\theta)}{2} \][/tex]
### Expression (c)
[tex]\[ (1 - \cos^2 \theta) (1 + \cot^2 \theta) \][/tex]
Rewrite using the Pythagorean identities:
[tex]\[ \sin^2 \theta \left( 1 + \frac{\cos^2 \theta}{\sin^2 \theta} \right) \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ \sin^2 \theta \left( \frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta} \right) \][/tex]
Since [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex], this becomes:
[tex]\[ \sin^2 \theta \left( \frac{1}{\sin^2 \theta} \right) = 1 \][/tex]
### Expression (d)
[tex]\[ \frac{\sec^2 \theta - \csc^2 \theta}{2} \][/tex]
Rewrite using the definitions:
[tex]\[ \frac{\frac{1}{\cos^2 \theta} - \frac{1}{\sin^2 \theta}}{2} \][/tex]
Combine the fractions:
[tex]\[ \frac{\frac{\sin^2 \theta - \cos^2 \theta}{\cos^2 \theta \sin^2 \theta}}{2} \][/tex]
Simplify the numerator using the identity [tex]\(\sin^2 \theta - \cos^2 \theta = -(\cos^2 \theta - \sin^2 \theta)\)[/tex]:
[tex]\[ \frac{- (\cos^2 \theta - \sin^2 \theta)}{2 \cos^2 \theta \sin^2 \theta} \][/tex]
Factor out the negative sign:
[tex]\[ - \frac{\cos^2 \theta - \sin^2 \theta}{2 \cos^2 \theta \sin^2 \theta} \][/tex]
Split the fraction:
[tex]\[ - \frac{\cos^2 \theta}{2 \cos^2 \theta \sin^2 \theta} + \frac{\sin^2 \theta}{2 \cos^2 \theta \sin^2 \theta} \][/tex]
Simplify each term:
[tex]\[ - \frac{1}{2 \sin^2 \theta} + \frac{1}{2 \cos^2 \theta} \][/tex]
Rewrite each fraction:
[tex]\[ - \frac{\csc^2 \theta}{2} + \frac{\sec^2 \theta}{2} \][/tex]
So the expression simplifies to:
[tex]\[ - \frac{\csc^2 \theta}{2} + \frac{\sec^2 \theta}{2} \][/tex]
### Summary
a) [tex]\[ \sec^2 \theta + \csc^2 \theta - \sec^2 \theta \cdot \csc^2 \theta = 0 \][/tex]
b) [tex]\[ (\sec \theta - \cos \theta)(\csc \theta - \sin \theta) = \frac{\sin(2\theta)}{2} \][/tex]
c) [tex]\[ (1 - \cos^2 \theta)(1 + \cot^2 \theta) = 1 \][/tex]
d) [tex]\[ \frac{\sec^2 \theta - \csc^2 \theta}{2} = - \frac{\csc^2 \theta}{2} + \frac{\sec^2 \theta}{2} \][/tex]
