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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
संबंधित प्रश्न
Prove that `\frac{\sin \theta -\cos \theta }{\sin \theta +\cos \theta }+\frac{\sin\theta +\cos \theta }{\sin \theta -\cos \theta }=\frac{2}{2\sin^{2}\theta -1}`
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:
(cos A + sin A)2 + (cos A – sin A)2 = 2
`sqrt((1+cos theta)/(1-cos theta)) + sqrt((1-cos theta )/(1+ cos theta )) = 2 cosec theta`
Prove that : `1 - (cos^2 θ)/(1 + sin θ) = sin θ`.
Prove the following identities:
`(1 - tan^2 θ)/(cot^2 θ - 1) = tan^2 θ`.
Prove the following identities.
cot θ + tan θ = sec θ cosec θ
Prove that cos2θ . (1 + tan2θ) = 1. Complete the activity given below.
Activity:
L.H.S = `square`
= `cos^2theta xx square .....[1 + tan^2theta = square]`
= `(cos theta xx square)^2`
= 12
= 1
= R.H.S
Given that sinθ + 2cosθ = 1, then prove that 2sinθ – cosθ = 2.
Prove the following trigonometry identity:
(sin θ + cos θ)(cosec θ – sec θ) = cosec θ ⋅ sec θ – 2 tan θ
