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प्रश्न
If tanθ = 1 then, find the value of
`(sinθ + cosθ)/(secθ + cosecθ)`
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उत्तर
tanθ = 1 ...(Given)
We know that, tan45° = 1
∴ θ = 45º
Now,
sinθ = sin 45º = `1/sqrt2`
cosθ = cos 45º = `1/sqrt2`
secθ = sec 45º = `sqrt2`
cosecθ = cosec 45º = `sqrt2`
∴ `(sinθ + cosθ)/(secθ + cosecθ)`
⇒ `(1/sqrt2 + 1/sqrt2)/(sqrt2 + sqrt2)`
⇒ `(2/sqrt2)/(2sqrt2)`
⇒ `cancel2/sqrt2 × 1/(cancel2sqrt2)`
⇒ `1/2`
∴ `(sinθ + cosθ)/(secθ + cosecθ) = 1/2`
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Proof: L.H.S. = (sec θ – cos θ) (cot θ + tan θ)
= `(1/square - cos θ) (square/square + square/square)` ......`[∵ sec θ = 1/square, cot θ = square/square and tan θ = square/square]`
= `((1 - square)/square) ((square + square)/(square square))`
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= `square/(square square)`
= tan θ.sec θ
= R.H.S.
∴ L.H.S. = R.H.S.
∴ (sec θ – cos θ) (cot θ + tan θ) = tan θ.sec θ
