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Question
`sqrt((1 - sin^2 θ)/(1 - cos^2 θ)` is equal to ______.
Options
tan θ
cot θ
sec θ
cos θ
MCQ
Fill in the Blanks
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Solution
`sqrt((1 - sin^2 θ)/(1 - cos^2 θ)` is equal to cot θ.
Explanation:
1. Use the Pythagorean identity sin2θ + cos2θ = 1.
So, 1 – sin2θ = cos2θ and 1 – cos2θ = sin2θ.
2. The expression becomes `sqrt(cos^2θ/sin^2θ)`
= `sqrt((cos θ/sin θ)^2)` ...(Recall `sqrt(a^2) = |a|`)
= |cot θ |
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Notes
The principal value is |cot θ|; the answer is written as cot θ in the choices option b, which is correct when cot θ ≥ 0 i.e., when sin and cos have the same sign.
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