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(1+tan theta)/(1-tan theta)+(1+cot theta)/(1-cot theta)

1+tan⁡θ1−tan⁡θ+1+cot⁡θ1−cot⁡θ \frac{1+\tan \theta}{1-\tan \theta}+\frac{1+\cot \theta}{1-\cot \theta}

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Full solution

Q. 1+tan⁡θ1−tan⁡θ+1+cot⁡θ1−cot⁡θ \frac{1+\tan \theta}{1-\tan \theta}+\frac{1+\cot \theta}{1-\cot \theta}
  1. Rewrite using identity: Step 11: Use the identity cot⁡(θ)=1tan⁡(θ)\cot(\theta) = \frac{1}{\tan(\theta)} to rewrite the second term.\newlineCalculation: 1+cot⁡θ1−cot⁡θ=1+1tan⁡(θ)1−1tan⁡(θ)\frac{1+\cot \theta}{1-\cot \theta} = \frac{1 + \frac{1}{\tan(\theta)}}{1 - \frac{1}{\tan(\theta)}}
  2. Simplify with common denominator: Step 22: Simplify the expression by finding a common denominator for the terms in the second fraction.\newlineCalculation: (1+1tan⁡(θ))/(1−1tan⁡(θ))=(tan⁡(θ)+1)/(tan⁡(θ)−1)(1 + \frac{1}{\tan(\theta)})/(1 - \frac{1}{\tan(\theta)}) = (\tan(\theta) + 1)/(\tan(\theta) - 1)
  3. Add fractions: Step 33: Add the two fractions together.\newlineCalculation: (1+tan⁡(θ)1−tan⁡(θ))+(tan⁡(θ)+1tan⁡(θ)−1)(\frac{1+\tan(\theta)}{1-\tan(\theta)}) + (\frac{\tan(\theta)+1}{\tan(\theta)-1})
  4. Find common denominator: Step 44: Find a common denominator for the entire expression.\newlineCalculation: [(1+tan⁡(θ))(tan⁡(θ)−1)+(tan⁡(θ)+1)(1−tan⁡(θ))(1−tan⁡(θ))(tan⁡(θ)−1)][\frac{(1+\tan(\theta))(\tan(\theta)-1) + (\tan(\theta)+1)(1-\tan(\theta))}{(1-\tan(\theta))(\tan(\theta)-1)}]
  5. Expand and simplify numerator: Step 55: Expand and simplify the numerator.\newlineCalculation: [\tan^\(2(\theta) - \tan(\theta) + \tan(\theta) - 11 + \tan(\theta) + 11 - \tan^22(\theta) + \tan(\theta)]/[(11-\tan(\theta))(\tan(\theta)−1-1)]

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