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प्रश्न
Prove the following identities:
(cosec A – sin A) (sec A – cos A) (tan A + cot A) = 1
Prove that:
(cosec A − sin A) (sec A – cos A) (tan A + cot A) = 1
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उत्तर
L.H.S. = (cosec A – sin A) (sec A – cos A) (tan A + cot A)
= `(1/sinA - sinA)(1/cosA - cosA)(1/tanA + tanA)`
= `((1 - sin^2A)/sinA)((1 - cos^2A)/cosA)(sinA/cosA + cosA/sinA)`
= `(cos^2A/sinA)(sin^2A/cosA)((sin^2A + cos^2A)/(sinA.cosA))`
= `(cos^2A/sinA)(sin^2A/cosA)((1)/(sinA.cosA))`
= `(cos^2A sin^2A)/((sinA .cosA)(sinA.cosA ))`
= `(cos^2A sin^2A)/(sin^2A cos^2A)`
= 1
= R.H.S.
संबंधित प्रश्न
Prove the following trigonometric identities.
`tan A/(1 + tan^2 A)^2 + cot A/((1 + cot^2 A)) = sin A cos A`
Prove the following identities:
`(sec A - 1)/(sec A + 1) = (1 - cos A)/(1 + cos A)`
Prove the following identities:
`sinA/(1 - cosA) - cotA = cosecA`
If (cosec θ – sin θ) = a3 and (sec θ – cos θ) = b3, prove that a2b2(a2 + b2) = 1.
Prove the following identity :
`(secA - 1)/(secA + 1) = (1 - cosA)/(1 + cosA)`
Without using trigonometric identity , show that :
`tan10^circ tan20^circ tan30^circ tan70^circ tan80^circ = 1/sqrt(3)`
Prove the following identities.
`sqrt((1 + sin theta)/(1 - sin theta)` = sec θ + tan θ
Complete the following activity to prove:
cotθ + tanθ = cosecθ × secθ
Activity: L.H.S. = cotθ + tanθ
= `cosθ/sinθ + square/cosθ`
= `(square + sin^2theta)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ....... ∵ `square`
= `1/sinθ xx 1/cosθ`
= `square xx secθ`
∴ L.H.S. = R.H.S.
Which of the following is true for all values of θ (0° ≤ θ ≤ 90°)?
Prove that `(1 + tan^2 A)/(1 + cot^2 A)` = sec2 A – 1
