Question

Prove that :- √{(sec θ – 1)/(sec θ + 1)} = cosec θ - cot θ.

Answer 1

SOLUTION

L.H.S

= √{(sec θ – 1)/(sec θ + 1)}

• [multiply numerator and denominator by √(sec θ - l)]

= √[{(sec θ - 1) (sec θ - 1)}/{(sec θ + 1) (sec θ - 1)}]

= √{(sec θ - 1)²/(sec²θ - 1)}

• [sec²θ = 1 + tan²θ ⇒ sec²θ - 1 = tan²θ]

=√{(sec θ - 1)²/tan²θ}

= (sec θ – 1)/tan θ

= (sec θ/tan θ) – (1/tan θ)

= {(1/cos θ)/(sin θ/cos θ)} - cot θ

= {(1/cos θ) × (cos θ/sin θ)} - cot θ

= (1/sin θ) - cot θ

= cosec θ - cot θ

= R.H.S

Hence Proved ✓✓

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