Advertisements
Advertisements
प्रश्न
Prove that `(cot A)/(1 - tan A) + (tan A)/(1 - cot A) = 1 + tan A + cot A = sec A . "cosec" A + 1`.
Advertisements
उत्तर
`(cot A)/(1 - tan A) + (tan A)/(1 - cot A)`
= `((cos A)/(sin A))/(1 - (sin A)/(cos A)) + ((sin A)/(cos A))/(1 - (cos A)/(sin A))`
= `((cos A)/(sin A))/((cos A - sin A)/(cos A)) + ((sin A)/(cos A))/((sin A - cos A)/(sin A))`
= `(cos A)/(sin A) xx (cos A)/(cos A - sin A) + (sin A)/(cos A) xx (sin A)/(sin A - cos A)`
= `(cos^2A)/(sin A(cos A - sin A)) + (sin^2A)/(cos A(sin A - cos A))`
= `1/(sin A - cos A) ((-cos^3A + sin^3A)/(sin A cos A))`
= `1/(sin A - cos A)((sin^3A - cos^3A)/(sin A cos A))`
= `1/(sin A - cos A) xx ((sin A - cos A)(sin^2A + sin A cos A + cos^2A))/(sin A cos A)` ...[∵ a3 – b3 = (a – b)(a2 + ab + b2)]
= `(sin^2A + sin A cos A + cos^2A)/(sin A cos A)` ...(i)
= `(1 + sin A cos A)/(sin A cos A)` ...[∵ sin2A + cos2A = 1]
= `1/(sin A cos A) + (sin A cos A)/(sin A cos A)`
= cosec A sec A + 1 ...(ii)
`(cot A)/(1 - tan A) + (tan A)/(1 - cot A)`
= `(sin^2A + sin A cos A + cos^2A)/(sin A cos A)` ...[From (i)]
= `(sin^2A)/(sin A cos A) + (sin A cos A)/(sin A cos A) + (cos^2A)/(sin A cos A)`
= `(sin A)/(cos A) + 1 + (cos A)/(sin A)`
= tan A + 1 + cot A ...(iii)
From (ii) and (iii), we get
`(cot A)/(1 - tan A) + (tan A)/(1 - cot A) = 1 + tan A + cot A = sec A . "cosec" A + 1`
संबंधित प्रश्न
Prove the following trigonometric identities.
`cos theta/(1 + sin theta) = (1 - sin theta)/cos theta`
Prove the following trigonometric identities.
if x = a cos^3 theta, y = b sin^3 theta` " prove that " `(x/a)^(2/3) + (y/b)^(2/3) = 1`
Prove the following identities:
`1 - cos^2A/(1 + sinA) = sinA`
Show that : tan 10° tan 15° tan 75° tan 80° = 1
` tan^2 theta - 1/( cos^2 theta )=-1`
`sec theta (1- sin theta )( sec theta + tan theta )=1`
`(sec theta -1 )/( sec theta +1) = ( sin ^2 theta)/( (1+ cos theta )^2)`
` (sin theta - cos theta) / ( sin theta + cos theta ) + ( sin theta + cos theta ) / ( sin theta - cos theta ) = 2/ ((2 sin^2 theta -1))`
Prove the following identities:
`(cot^2 theta (sec theta - 1))/((1 + sin theta)) + (sec^2 theta(sin theta - 1))/((1 + sec theta)) = 0`
If (cot θ + tan θ) = m and (sec θ – cos θ) = n, prove that `(m^2 n)^(2//3) - (mn^2)^(2//3) = 1`.
If `cos theta = 2/3 , " write the value of" (4+4 tan^2 theta).`
\[\frac{\tan \theta}{\sec \theta - 1} + \frac{\tan \theta}{\sec \theta + 1}\] is equal to
Prove the following identity :
`sec^4A - sec^2A = sin^2A/cos^4A`
Prove the following identity :
`sin^8θ - cos^8θ = (sin^2θ - cos^2θ)(1 - 2sin^2θcos^2θ)`
Evaluate:
`(tan 65^circ)/(cot 25^circ)`
Prove that : `(sin(90° - θ) tan(90° - θ) sec (90° - θ))/(cosec θ. cos θ. cot θ) = 1`
Prove the following identities:
`(1 - tan^2 θ)/(cot^2 θ - 1) = tan^2 θ`.
The value of sin2θ + `1/(1 + tan^2 theta)` is equal to
Prove that `(1 + sin B)/(cos B) + (cos B)/(1 + sin B) = 2 sec B`.
