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Q. simplify
- Recall Trigonometric Identities: Let's start by recalling the trigonometric identities that relate cotangent and cosine: and , so .Therefore, .Now let's rewrite the expression using these identities.
- Substitute and Rewrite Expression: Substitute with in the expression:$(\cot^\(2\)(x) \cdot \cos^\(2\)(x)) / (\cot^\(2\)(x) - \cos^\(2\)(x)) = \left(\frac{\cos^\(2\)(x)}{\sin^\(2\)(x)} \cdot \cos^\(2\)(x)\right) / \left(\frac{\cos^\(2\)(x)}{\sin^\(2\)(x)} - \cos^\(2\)(x)\right).
- Simplify Numerator and Denominator: Now, let's simplify the numerator and the denominator separately:\(\newline\)Numerator: \((\cos^2(x)/\sin^2(x)) \cdot \cos^2(x) = \cos^4(x)/\sin^2(x)\).\(\newline\)Denominator: \((\cos^2(x)/\sin^2(x)) - \cos^2(x) = \cos^2(x)/\sin^2(x) - (\cos^2(x) \cdot \sin^2(x))/\sin^2(x) = (\cos^2(x) - \cos^2(x) \cdot \sin^2(x))/\sin^2(x)\).
- Further Simplify Denominator: Simplify the denominator further: \(\newline\)\((\cos^2(x) - \cos^2(x) \sin^2(x))/\sin^2(x) = \cos^2(x)(1 - \sin^2(x))/\sin^2(x)\).\(\newline\)Recall the Pythagorean identity: \(\sin^2(x) + \cos^2(x) = 1\), so \(1 - \sin^2(x) = \cos^2(x)\).
- Apply Pythagorean Identity: Substitute \(1 - \sin^2(x)\) with \(\cos^2(x)\) in the denominator:\(\newline\)\(\cos^2(x)(1 - \sin^2(x))/\sin^2(x) = \cos^2(x) \cdot \cos^2(x)/\sin^2(x) = \cos^4(x)/\sin^2(x)\).
- Final Simplification: Now we have the same expression in both the numerator and the denominator: \(\newline\)\((\frac{\cos^4(x)}{\sin^2(x)}) / (\frac{\cos^4(x)}{\sin^2(x)})\).\(\newline\)This simplifies to \(1\), because any non-zero number divided by itself is \(1\).
