Advertisements
Advertisements
प्रश्न
Prove that `"cot A"/(1 - cot"A") + "tan A"/(1 - tan "A")` = – 1
उत्तर
L.H.S = `"cot A"/(1 - cot"A") + "tan A"/(1 - tan "A")`
= `"cot A"/(1 - 1/(tan"A")) + "tan A"/(1 - tan "A")`
= `"cot A"/((tan "A" - 1)/(tan "A")) + "tan A"/(1 - tan "A")`
= `"cot A tan A"/(tan "A" - 1) + "tan A"/(1 - tan "A")`
= `1/(tan "A" - 1) + "tan A"/(1 - tan "A")` ......[∵ cot A tan A = 1]
= `- 1/(1 - tan "A") + "tan A"/(1 - tan "A")`
= `- (1/(1 -tan "A") - "tan A"/(1- tan "A"))`
= `-((1 - tan "A")/(1 - tan "A"))`
= – 1
= R.H.S
∴ `"cot A"/(1 - cot"A") + "tan A"/(1 - tan "A")` = – 1
APPEARS IN
संबंधित प्रश्न
Prove the following trigonometric identities.
`tan theta + 1/tan theta = sec theta cosec theta`
Prove the following identities:
`1/(sinA + cosA) + 1/(sinA - cosA) = (2sinA)/(1 - 2cos^2A)`
If x cos A + y sin A = m and x sin A – y cos A = n, then prove that : x2 + y2 = m2 + n2
If x = r sin A cos B, y = r sin A sin B and z = r cos A, then prove that : x2 + y2 + z2 = r2
If tan A = n tan B and sin A = m sin B, prove that:
`cos^2A = (m^2 - 1)/(n^2 - 1)`
`sin^6 theta + cos^6 theta =1 -3 sin^2 theta cos^2 theta`
If `cos theta = 2/3 , "write the value of" ((sec theta -1))/((sec theta +1))`
Prove that:
`"tan A"/(1 + "tan"^2 "A")^2 + "Cot A"/(1 + "Cot"^2 "A")^2 = "sin A cos A"`.
Prove the following identity :
`sqrt((1 - cosA)/(1 + cosA)) = sinA/(1 + cosA)`
Prove the following identity :
`(secθ - tanθ)^2 = (1 - sinθ)/(1 + sinθ)`
Prove the following identity :
`(sec^2θ - sin^2θ)/tan^2θ = cosec^2θ - cos^2θ`
Prove that `(tan θ)/(cot(90° - θ)) + (sec (90° - θ) sin (90° - θ))/(cosθ. cosec θ) = 2`.
If `cos theta/(1 + sin theta) = 1/"a"`, then prove that `("a"^2 - 1)/("a"^2 + 1)` = sin θ
Prove that cot2θ × sec2θ = cot2θ + 1
Prove that `(cos^2theta)/(sintheta) + sintheta` = cosec θ
To prove cot θ + tan θ = cosec θ × sec θ, complete the activity given below.
Activity:
L.H.S = `square`
= `square/sintheta + sintheta/costheta`
= `(cos^2theta + sin^2theta)/square`
= `1/(sintheta*costheta)` ......`[cos^2theta + sin^2theta = square]`
= `1/sintheta xx 1/square`
= `square`
= R.H.S
Prove that
sec2A – cosec2A = `(2sin^2"A" - 1)/(sin^2"A"*cos^2"A")`
Prove the following:
`sintheta/(1 + cos theta) + (1 + cos theta)/sintheta` = 2cosecθ
(tan θ + 2)(2 tan θ + 1) = 5 tan θ + sec2θ.
(sec2 θ – 1) (cosec2 θ – 1) is equal to ______.
