Consider the following statements: 1. (sec 2θ - 1) (1 - cosec 2θ) = 1 2. sin θ (1 + cos θ) -1 + (1 + cos θ) (sin θ) -1 = 2 cosec θ
2 only
Evaluating Trigonometric Statements
We are given two trigonometric statements and asked to determine which one is correct. To do this, we will analyze each statement individually and simplify the expressions to see if the left-hand side (LHS) equals the right-hand side (RHS).
Analyzing Statement 1: (sec 2θ - 1) (1 - cosec 2θ) = 1
Let's examine the first statement:
\((\sec^2 \theta - 1)(1 - \operatorname{cosec}^2 \theta) = 1\)
We can use fundamental trigonometric identities to simplify the terms in the parentheses.
- Recall the Pythagorean identity: \(\sec^2 \theta - \tan^2 \theta = 1\). Rearranging this, we get \(\sec^2 \theta - 1 = \tan^2 \theta\).
- Recall the Pythagorean identity: \(\operatorname{cosec}^2 \theta - \cot^2 \theta = 1\). Rearranging this, we get \(1 - \operatorname{cosec}^2 \theta = -\cot^2 \theta\).
Now, substitute these simplified expressions back into the LHS of Statement 1:
LHS = \((\sec^2 \theta - 1)(1 - \operatorname{cosec}^2 \theta)\)
LHS = \((\tan^2 \theta)(-\cot^2 \theta)\)
We also know that \(\cot \theta = \frac{1}{\tan \theta}\), so \(\cot^2 \theta = \frac{1}{\tan^2 \theta}\).
Substitute this into the LHS expression:
LHS = \((\tan^2 \theta)\left(-\frac{1}{\tan^2 \theta}\right)\)
LHS = \(-1\)
The RHS of Statement 1 is \(1\).
Since \(-1 \neq 1\), Statement 1 is incorrect.
Analyzing Statement 2: sin θ (1 + cos θ) -1 + (1 + cos θ) (sin θ) -1 = 2 cosec θ
Let's examine the second statement. We can rewrite the terms with negative exponents as fractions:
\(\frac{\sin \theta}{1 + \cos \theta} + \frac{1 + \cos \theta}{\sin \theta} = 2 \operatorname{cosec} \theta\)
To simplify the LHS, find a common denominator, which is \((1 + \cos \theta)\sin \theta\):
LHS = \(\frac{\sin \theta \cdot \sin \theta}{(1 + \cos \theta)\sin \theta} + \frac{(1 + \cos \theta)(1 + \cos \theta)}{\sin \theta (1 + \cos \theta)}\)
LHS = \(\frac{\sin^2 \theta + (1 + \cos \theta)^2}{(1 + \cos \theta)\sin \theta}\)
Expand the term \((1 + \cos \theta)^2\) in the numerator:
\((1 + \cos \theta)^2 = 1^2 + 2(1)(\cos \theta) + \cos^2 \theta = 1 + 2 \cos \theta + \cos^2 \theta\)
Substitute this back into the numerator:
Numerator = \(\sin^2 \theta + 1 + 2 \cos \theta + \cos^2 \theta\)
Rearrange the terms and use the identity \(\sin^2 \theta + \cos^2 \theta = 1\):
Numerator = \((\sin^2 \theta + \cos^2 \theta) + 1 + 2 \cos \theta\)
Numerator = \(1 + 1 + 2 \cos \theta\)
Numerator = \(2 + 2 \cos \theta\)
Numerator = \(2(1 + \cos \theta)\)
Now substitute the simplified numerator back into the LHS expression:
LHS = \(\frac{2(1 + \cos \theta)}{(1 + \cos \theta)\sin \theta}\)
Assuming \(1 + \cos \theta \neq 0\), we can cancel the term \((1 + \cos \theta)\) from the numerator and denominator:
LHS = \(\frac{2}{\sin \theta}\)
Recall that \(\operatorname{cosec} \theta = \frac{1}{\sin \theta}\).
LHS = \(2 \cdot \frac{1}{\sin \theta}\)
LHS = \(2 \operatorname{cosec} \theta\)
The RHS of Statement 2 is \(2 \operatorname{cosec} \theta\).
Since LHS = RHS, Statement 2 is correct.
Conclusion on Trigonometric Statements
Based on our analysis:
- Statement 1: \((sec^2 \theta - 1)(1 - cosec^2 \theta) = 1\) is incorrect, as it simplifies to \(-1 = 1\).
- Statement 2: \(sin \theta (1 + cos \theta)^{-1} + (1 + cos \theta) (sin \theta)^{-1} = 2 cosec \theta\) is correct, as it simplifies to \(2 \operatorname{cosec} \theta = 2 \operatorname{cosec} \theta\).
Therefore, only Statement 2 is correct.
| Statement | Original Expression | Simplified LHS | RHS | Correctness |
|---|---|---|---|---|
| 1 | \((\sec^2 \theta - 1)(1 - \operatorname{cosec}^2 \theta) = 1\) | \(-1\) | \(1\) | Incorrect |
| 2 | \(\frac{\sin \theta}{1 + \cos \theta} + \frac{1 + \cos \theta}{\sin \theta} = 2 \operatorname{cosec} \theta\) | \(2 \operatorname{cosec} \theta\) | \(2 \operatorname{cosec} \theta\) | Correct |
Revision Table: Key Trigonometric Identities
| Identity Type | Identity |
|---|---|
| Reciprocal Identities | \(\sin \theta = \frac{1}{\operatorname{cosec} \theta}, \cos \theta = \frac{1}{\sec \theta}, \tan \theta = \frac{1}{\cot \theta}\) |
| Quotient Identities | \(\tan \theta = \frac{\sin \theta}{\cos \theta}, \cot \theta = \frac{\cos \theta}{\sin \theta}\) |
| Pythagorean Identities | \(\sin^2 \theta + \cos^2 \theta = 1\) |
| \(\tan^2 \theta + 1 = \sec^2 \theta \implies \sec^2 \theta - 1 = \tan^2 \theta\) | |
| \(\cot^2 \theta + 1 = \operatorname{cosec}^2 \theta \implies \operatorname{cosec}^2 \theta - 1 = \cot^2 \theta\) |
Additional Information: Steps for Verifying Trigonometric Identities
When asked to verify a trigonometric identity, here are some general steps you can follow:
- Start with one side: It's often easier to start with the more complex side of the identity and simplify it until it matches the other side.
- Use fundamental identities: Substitute expressions using reciprocal, quotient, or Pythagorean identities to simplify.
- Convert to sine and cosine: If stuck, rewrite all trigonometric functions in terms of sine and cosine. This often helps in finding common denominators and combining terms.
- Find a common denominator: If adding or subtracting fractions, find a common denominator to combine the terms into a single fraction.
- Factor expressions: Look for opportunities to factor algebraic expressions, such as differences of squares or perfect square trinomials.
- Multiply by a conjugate: Sometimes, multiplying the numerator and denominator by the conjugate of an expression (like \(1 + \cos \theta\) or \(1 - \sin \theta\)) can help simplify the expression, especially when dealing with fractions involving binomials.
- Simplify carefully: Perform algebraic manipulations correctly and simplify expressions step-by-step.
- Do not assume the identity is true: Do not perform the same operation on both sides of the equation unless you are proving it is an identity, not verifying it. Stick to transforming one side into the other.
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