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प्रश्न
Prove the following trigonometric identities.
`(1 + sin θ)/cos θ+ cos θ/(1 + sin θ) = 2 sec θ`
Prove the following:
`(1 + sin θ)/cos θ+ cos θ/(1 + sin θ) = 2 sec θ`
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उत्तर १
We have to prove `(1 + sin θ)/cos θ + cos θ/(1 + sin θ) = 2 sec θ`
We know that, `sin^2 θ + cos^2 θ = 1`
Multiplying the denominator and numerator of the second term by (1 − sin θ), we have
= `(1 + sin θ)/cos θ + cos θ/(1 + sin θ)`
`(1 + sin θ)/cos θ = (cos θ(1 - sin θ))/((1 + sin θ)(1 - sin θ))`
`(1 + sin θ)/cos θ = (cos θ (1 - sin θ))/(1-sin θ)`
= `(1 + sin θ)/cos θ + (cos θ(1 - sin θ))/cos^2 θ`
= `(1 + sin θ)/cos θ + (1 - sin θ)/cos θ`
= `(1 + sin θ + 1 - sin θ)/cos θ`
`= 2/cos θ`
= 2 sec θ
उत्तर २
LHS = `(1 + sin θ)/cos θ + cos θ/(1 + sin θ)`
= `(( 1 + sin θ)^2 + cos^2 θ)/(cos θ(1 + sin θ))`
= `(1 + sin^2 θ + 2 sin θ + cos^2 θ)/(cos θ(1 + sin θ ))`
= `(1 + (sin^2θ + cos^2 θ) + 2 sin θ)/(cos θ(1 + sin θ))`
= `(1 + 1 + 2sin θ)/(cos θ(1 + sin θ))`
= `(2(1 + sin θ))/(cos θ(1 + sin θ))`
= 2 sec θ
Hence proved.
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