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Prove step by step that (sec⁡sec⁡x)÷(tan⁡tan⁡x)−(tan⁡tan⁡x)÷(1+sec⁡sec⁡x)(\sec \sec x) \div (\tan \tan x) - (\tan \tan x) \div (1 + \sec\sec x) gives: cot⁡cot⁡x\cot \cot x

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Full solution

Q. Prove step by step that (sec⁡sec⁡x)÷(tan⁡tan⁡x)−(tan⁡tan⁡x)÷(1+sec⁡sec⁡x)(\sec \sec x) \div (\tan \tan x) - (\tan \tan x) \div (1 + \sec\sec x) gives: cot⁡cot⁡x\cot \cot x
  1. Rewrite sec⁡sec⁡x\sec \sec x: Rewrite sec⁡sec⁡x\sec \sec x as 1cos⁡(cos⁡x)\frac{1}{\cos(\cos x)} and tan⁡tan⁡x\tan \tan x as sin⁡(sin⁡x)cos⁡(cos⁡x)\frac{\sin(\sin x)}{\cos(\cos x)}.1cos⁡(cos⁡x)÷sin⁡(sin⁡x)cos⁡(cos⁡x)−sin⁡(sin⁡x)cos⁡(cos⁡x)÷(1+1cos⁡(cos⁡x))\frac{1}{\cos(\cos x)} \div \frac{\sin(\sin x)}{\cos(\cos x)} - \frac{\sin(\sin x)}{\cos(\cos x)} \div \left(1 + \frac{1}{\cos(\cos x)}\right)
  2. Simplify division: Simplify the division by multiplying by the reciprocal.\newline(1cos⁡(cos⁡x))×(cos⁡(cos⁡x)sin⁡(sin⁡x))−(sin⁡(sin⁡x)cos⁡(cos⁡x))×(cos⁡(cos⁡x)1+1cos⁡(cos⁡x))(\frac{1}{\cos(\cos x)}) \times (\frac{\cos(\cos x)}{\sin(\sin x)}) - (\frac{\sin(\sin x)}{\cos(\cos x)}) \times (\frac{\cos(\cos x)}{1 + \frac{1}{\cos(\cos x)}})
  3. Simplify expressions: Simplify the expressions. cos⁡(cos⁡x)sin⁡(sin⁡x)\frac{\cos(\cos x)}{\sin(\sin x)} - sin⁡(sin⁡x)cos⁡(cos⁡x)cos⁡(cos⁡x)+1\frac{\sin(\sin x)\cos(\cos x)}{\cos(\cos x) + 1}
  4. Combine terms: Combine the terms over a common denominator.\newline(cos⁡(cos⁡x)sin⁡(sin⁡x))×(cos⁡(cos⁡x)+1cos⁡(cos⁡x)+1)−(sin⁡(sin⁡x)cos⁡(cos⁡x)cos⁡(cos⁡x)+1)(\frac{\cos(\cos x)}{\sin(\sin x)}) \times (\frac{\cos(\cos x) + 1}{\cos(\cos x) + 1}) - (\frac{\sin(\sin x)\cos(\cos x)}{\cos(\cos x) + 1})
  5. Simplify numerator: Simplify the numerator. cos⁡2(cos⁡x)+cos⁡(cos⁡x)−sin⁡(sin⁡x)cos⁡(cos⁡x)cos⁡(cos⁡x)+1\frac{\cos^2(\cos x) + \cos(\cos x) - \sin(\sin x)\cos(\cos x)}{\cos(\cos x) + 1}
  6. Factor out cos: Factor out cos⁡(cos⁡x)\cos(\cos x) in the numerator.cos⁡(cos⁡x)(cos⁡(cos⁡x)+1−sin⁡(sin⁡x))cos⁡(cos⁡x)+1\frac{\cos(\cos x)(\cos(\cos x) + 1 - \sin(\sin x))}{\cos(\cos x) + 1}
  7. Cancel terms: Cancel out the cos⁡(cos⁡x)+1\cos(\cos x) + 1 terms in the numerator and denominator.cos⁡(cos⁡x)sin⁡(sin⁡x)\frac{\cos(\cos x)}{\sin(\sin x)}
  8. Rewrite as cot⁡\cot: Rewrite cos⁡(cos⁡x)sin⁡(sin⁡x)\frac{\cos(\cos x)}{\sin(\sin x)} as cot⁡(cot⁡x)\cot(\cot x).cot⁡(cot⁡x)\cot(\cot x)

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