Consider the following statements :
I. \(\tan 50^\circ - \cot 50^\circ\) is positive
II. \(\cot 25^\circ - \tan 25^\circ\) is negative
Which of the statements is/are correct?
Trigonometric Statement Analysis
This solution analyzes two statements involving the trigonometric functions tangent (tan) and cotangent (cot) to determine their correctness based on the sign (positive or negative) of the given expressions. Key trigonometric identities and the behavior of trigonometric functions are used.
Statement I: Evaluating \(tan 50° - cot 50°\)
We need to determine if the expression \(\tan 50^\circ - \cot 50^\circ\) is positive.
- Rewrite using co-function identity: Recall the identity \(\cot \theta = \tan(90^\circ - \theta)\). Applying this to \(\cot 50^\circ\), we get: \( \cot 50^\circ = \tan(90^\circ - 50^\circ) = \tan 40^\circ \)
- Simplify the expression: Substituting this back, the expression becomes: \( \tan 50^\circ - \tan 40^\circ \)
- Analyze tangent function behavior: The tangent function (\(\tan x\)) is strictly increasing in the interval \((0^\circ, 90^\circ)\). This means for any two angles \(a\) and \(b\) in this interval, if \(a > b\), then \(\tan a > \tan b\).
- Compare angles: In our expression, we have \(50^\circ > 40^\circ\). Both angles are within the \((0^\circ, 90^\circ)\) interval.
- Determine the sign: Since \(50^\circ > 40^\circ\), it follows that \(\tan 50^\circ > \tan 40^\circ\). Therefore, their difference, \(\tan 50^\circ - \tan 40^\circ\), must be a positive value.
- Conclusion for Statement I: Statement I, which claims the expression is positive, is correct.
Statement II: Evaluating \(cot 25° - tan 25°\)
We need to determine if the expression \(\cot 25^\circ - \tan 25^\circ\) is negative.
Method 1: Using Cofunction Identity and Tangent Behavior
- Rewrite using co-function identity: Using \(\cot \theta = \tan(90^\circ - \theta)\), we get: \( \cot 25^\circ = \tan(90^\circ - 25^\circ) = \tan 65^\circ \)
- Simplify the expression: The expression becomes: \( \tan 65^\circ - \tan 25^\circ \)
- Analyze and compare angles: Since \(65^\circ > 25^\circ\) and the tangent function is increasing in the first quadrant, we know \(\tan 65^\circ > \tan 25^\circ\).
- Determine the sign: Therefore, the difference \(\tan 65^\circ - \tan 25^\circ\) is positive.
- Conclusion for Statement II (Method 1): The statement claims the result is negative, but our analysis shows it is positive. Thus, Statement II is incorrect.
Method 2: Using Sine and Cosine Identities
- Rewrite in terms of sine and cosine: \( \cot 25^\circ - \tan 25^\circ = \frac{\cos 25^\circ}{\sin 25^\circ} - \frac{\sin 25^\circ}{\cos 25^\circ} \)
- Combine fractions: Find a common denominator, \(\sin 25^\circ \cos 25^\circ\): \( = \frac{\cos^2 25^\circ - \sin^2 25^\circ}{\sin 25^\circ \cos 25^\circ} \)
- Apply double-angle identities: Use \(\cos(2\theta) = \cos^2 \theta - \sin^2 \theta\) and \(\sin(2\theta) = 2 \sin \theta \cos \theta\).
- The numerator is \(\cos(2 \times 25^\circ) = \cos 50^\circ\).
- The denominator is \(\frac{1}{2} \sin(2 \times 25^\circ) = \frac{1}{2} \sin 50^\circ\).
- Simplify the expression: \( = \frac{\cos 50^\circ}{\frac{1}{2} \sin 50^\circ} = 2 \frac{\cos 50^\circ}{\sin 50^\circ} = 2 \cot 50^\circ \)
- Analyze the sign: The angle \(50^\circ\) is in the first quadrant (\(0^\circ < 50^\circ < 90^\circ\)), where the cotangent function is positive. Thus, \(2 \cot 50^\circ\) is positive.
- Conclusion for Statement II (Method 2): This method also confirms that the expression is positive, contradicting the statement that it is negative. Statement II is incorrect.
Final Conclusion
After analyzing both statements:
- Statement I is correct, as \(\tan 50^\circ - \cot 50^\circ\) results in a positive value.
- Statement II is incorrect, as \(\cot 25^\circ - \tan 25^\circ\) results in a positive value, not negative.
Therefore, only Statement I is correct.
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