Checkout JEE MAINS 2022 Question Paper Analysis : Checkout JEE MAINS 2022 Question Paper Analysis :

(√2 - sin α - cos α)/(sin α - cos α) =

1) sec(α/2 – π/8)

2) cos(π/8 – α/2)

3) tan(α/2 – π/8)

4) cot(α/2 – π/2)

Answer: (3) tan(α/2 – π/8)

Solution:

(√2 – sin α – cos α)/(sin α – cos α)

[√2 – √2{(1/√2) sin α + (1/√2)cos α}]/[√2 {(1/√2) sin α – (1/√2) cos α}]

= [√2 – √2{sin π/4 sin α + cos π/4 cos α}]/[√2 {cos π/4 sin α – sin π/4 cos α}]

= [√2 – √2 cos(α – π/4)]/ [√2 sin( α- π/4)]

Let us assume θ = α – π/4….(i)

= (√2 – √2 cos θ)/(√2 sin θ)

= √2(1 – cos θ)/(√2 sin θ)

= (1 – cos θ)/sin θ

= [2 sin2(θ/2)]/[2 sin θ/2 cos θ/2]

= (sin θ/2)/ (cos θ/2)

= tan θ/2

= tan(α/2 – π/8) {from (i)}

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