What is the value of \(\rm \sqrt{\frac{1+\tan^2 \theta}{1+\cot ^2\theta}} \)?

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DFCCIL Executive CBT1 Civil 25 Aug 2023 Shift 3 Official Paper
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  1. tan θ cot θ 
  2. cot θ 
  3. 1
  4. tan θ

Answer (Detailed Solution Below)

Option 4 : tan θ
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Detailed Solution

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Given:

Expression: √[(1 + tan²θ) / (1 + cot²θ)]

Formula used:

1 + tan²θ = sec²θ

1 + cot²θ = csc²θ

Calculation:

√[(1 + tan²θ) / (1 + cot²θ)]

⇒ √[(sec²θ) / (csc²θ)]

⇒ √[(1 / cos²θ) / (1 / sin²θ)]

⇒ √[(sin²θ / cos²θ)]

⇒ √(tan²θ) = tanθ

∴ Value = tanθ

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