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प्रश्न
If sec θ = `5/4 ` show that `((sin θ - 2 cos θ))/(( tan θ - cot θ)) = 12/7`
उत्तर
We have ,
sec θ = `5/4`
`⇒ 1/(cos θ) = 5/4`
`⇒ cos θ = 4/5`
Also,
`sin^2 θ = 1-cos^2 θ`
`= 1-(4/5)^2`
`= 1-16/25`
=`9/25`
`⇒ sin θ = 3/5`
Now ,
LHS = `((sin θ -2 cos θ))/((tan θ - cot θ))`
= `((sin θ - 2 cos θ))/(((sin θ cos θ)/(cos θ sin θ))`
=` ((sin θ - 2 - cos θ))/(((sin^2 θ- cos^2 θ)/(sin θ cosθ)))`
`(sinθ cosθ (sinθ-2cosθ))/((sin^2θ-cos^2 θ))`
`=(3/5xx4/5(3/5-2xx4/5))/((3/5)^2-(4/5)^2`
=`(12/25(3/5-8/5))/((9/25-16/25))`
`= (12/25xx((-5)/5))/((-7/25))`
=`12/7`
= RHS
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