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प्रश्न
Prove the following identities:
(1 + tan A + sec A) (1 + cot A – cosec A) = 2
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उत्तर
(1 + tan A + sec A) (1 + cot A – cosec A)
= 1 + cot A – cosec A + tan A + 1 – sec A + sec A + cosec A – cosec A sec A
= `2 + cosA/sinA + sinA/cosA - 1/(sinAcosA)`
= `2 + (cos^2A + sin^2A)/(sinAcosA) - 1/(sinAcosA)`
= `2 + 1/(sinAcosA) - 1/(sinAcosA)`
= 2
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Show that, cotθ + tanθ = cosecθ × secθ
Solution :
L.H.S. = cotθ + tanθ
= `cosθ/sinθ + sinθ/cosθ`
= `(square + square)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ............... `square`
= `1/sinθ xx 1/square`
= cosecθ × secθ
L.H.S. = R.H.S
∴ cotθ + tanθ = cosecθ × secθ
