What is (1 + cot θ - cosec θ)(1 + tan θ + sec θ) equal to?
2
Understanding the Trigonometric Expression
The question asks us to simplify and find the value of the expression \((1 + \cot \theta - \operatorname{cosec} \theta)(1 + \tan \theta + \sec \theta)\). This involves using fundamental trigonometric identities and algebraic manipulation.
To simplify expressions involving different trigonometric functions like \(\cot \theta\), \(\operatorname{cosec} \theta\), \(\tan \theta\), and \(\sec \theta\), it is often helpful to convert them into their basic forms involving \(\sin \theta\) and \(\cos \theta\).
- We know that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
- We know that \(\operatorname{cosec} \theta = \frac{1}{\sin \theta}\)
- We know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- We know that \(\sec \theta = \frac{1}{\cos \theta}\)
Step-by-Step Simplification
Let's substitute these equivalent forms into the given expression:
The expression is: \((1 + \cot \theta - \operatorname{cosec} \theta)(1 + \tan \theta + \sec \theta)\)
Substitute the \(\sin \theta\) and \(\cos \theta\) forms:
\(\left(1 + \frac{\cos \theta}{\sin \theta} - \frac{1}{\sin \theta}\right)\left(1 + \frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta}\right)\)
Now, let's simplify the terms inside each set of parentheses by finding a common denominator:
For the first set of parentheses, the common denominator is \(\sin \theta\):
\(1 + \frac{\cos \theta}{\sin \theta} - \frac{1}{\sin \theta} = \frac{\sin \theta}{\sin \theta} + \frac{\cos \theta}{\sin \theta} - \frac{1}{\sin \theta} = \frac{\sin \theta + \cos \theta - 1}{\sin \theta}\)
For the second set of parentheses, the common denominator is \(\cos \theta\):
\(1 + \frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta} = \frac{\cos \theta}{\cos \theta} + \frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta} = \frac{\cos \theta + \sin \theta + 1}{\cos \theta}\)
Now, multiply these two simplified expressions:
\(\left(\frac{\sin \theta + \cos \theta - 1}{\sin \theta}\right)\left(\frac{\sin \theta + \cos \theta + 1}{\cos \theta}\right)\)
We can rewrite the numerators slightly to group the terms \((\sin \theta + \cos \theta)\):
\(\frac{(\sin \theta + \cos \theta) - 1}{\sin \theta} \times \frac{(\sin \theta + \cos \theta) + 1}{\cos \theta} = \frac{((\sin \theta + \cos \theta) - 1)((\sin \theta + \cos \theta) + 1)}{\sin \theta \cos \theta}\)
The numerator is in the form \((a-b)(a+b)\), where \(a = (\sin \theta + \cos \theta)\) and \(b = 1\). Using the algebraic identity \((a-b)(a+b) = a^2 - b^2\), we get:
Numerator \(= (\sin \theta + \cos \theta)^2 - 1^2\)
Expand \((\sin \theta + \cos \theta)^2\) using \((a+b)^2 = a^2 + b^2 + 2ab\):
\((\sin \theta + \cos \theta)^2 = \sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta\)
We know the fundamental trigonometric identity \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute this into the expanded term:
\((\sin \theta + \cos \theta)^2 = 1 + 2 \sin \theta \cos \theta\)
Now substitute this back into the numerator expression:
Numerator \(= (1 + 2 \sin \theta \cos \theta) - 1\)
Simplify the numerator:
Numerator \(= 1 + 2 \sin \theta \cos \theta - 1 = 2 \sin \theta \cos \theta\)
So, the entire expression becomes:
\(\frac{2 \sin \theta \cos \theta}{\sin \theta \cos \theta}\)
Assuming \(\sin \theta \cos \theta \neq 0\) (which means \(\theta\) is not a multiple of \(\frac{\pi}{2}\)), we can cancel out the term \(\sin \theta \cos \theta\) from the numerator and the denominator:
\(\frac{2 \cancel{\sin \theta \cos \theta}}{\cancel{\sin \theta \cos \theta}} = 2\)
Thus, the value of the expression \((1 + \cot \theta - \operatorname{cosec} \theta)(1 + \tan \theta + \sec \theta)\) is 2.
Revision Table: Key Trigonometric Identities
| Identity Type | Identity |
|---|---|
| Reciprocal Identities |
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| Quotient Identities |
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| Pythagorean Identities |
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Additional Information: Simplifying Trigonometric Expressions
When faced with simplifying complex trigonometric expressions, here are some general strategies:
- Convert to Sine and Cosine: As shown in this solution, expressing everything in terms of \(\sin \theta\) and \(\cos \theta\) is often the first step and can reveal opportunities for simplification.
- Look for Identities: Be familiar with reciprocal, quotient, and Pythagorean identities. They are crucial for transforming expressions.
- Factor and Expand: Use algebraic techniques like factoring, expanding, finding common denominators, and using difference of squares (\(a^2 - b^2\)) or perfect squares (\((a+b)^2\)) formulas.
- Simplify Fractions: Combine terms into a single fraction and look for common factors in the numerator and denominator to cancel.
- Work on One Side: If proving an identity, work on the more complicated side to transform it into the simpler side. For simplification problems like this one, work on the given expression until it is in its simplest form.
This problem specifically utilized converting to \(\sin \theta\) and \(\cos \theta\), finding common denominators, multiplying fractions, and applying the algebraic identity for difference of squares along with the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\).
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