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प्रश्न
Prove that: `1/(cosec"A" - cot"A") - 1/sin"A" = 1/sin"A" - 1/(cosec"A" + cot"A")`
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उत्तर
`= 1/(cosectheta-cottheta)xx (cosectheta+cottheta)/(cosectheta+cottheta)-1/sintheta`
`= (cosectheta+cottheta)/(cosec^2theta-cot^2theta) - 1/sintheta`
`cosectheta+cottheta - 1/sintheta`
`1/sintheta+costheta/sintheta - 1/sintheta`
`1/sintheta+(costheta-1)/sinthetaxx(costheta+1)/(costheta+1)`
`1/sintheta+(cos^2theta-1)/((1+costheta)sintheta)`
`1/sintheta-(1-cos^2theta)/(sintheta(1+costheta))`
`1/sintheta - (sin2theta)/(sintheta(1+costheta))`
`1/sintheta-sintheta/(1+costheta)`
`1/sintheta - (sintheta/sintheta)/(1/sintheta+costheta/sintheta)`
`1/sintheta-1/(cosectheta+cottheta)`= RHS
संबंधित प्रश्न
Prove the following trigonometric identities.
`1/(1 + sin A) + 1/(1 - sin A) = 2sec^2 A`
If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z = c tan θ, show that `x^2/a^2 + y^2/b^2 - x^2/c^2 = 1`
Prove the following identities:
`(1 - cosA)/sinA + sinA/(1 - cosA)= 2cosecA`
`sin theta/((cot theta + cosec theta)) - sin theta /( (cot theta - cosec theta)) =2`
Prove that secθ + tanθ =`(costheta)/(1-sintheta)`.
If \[\sin \theta = \frac{1}{3}\] then find the value of 2cot2 θ + 2.
Prove the following identity :
`(cotA - cosecA)^2 = (1 - cosA)/(1 + cosA)`
If sin θ + cos θ = `sqrt(3)`, then prove that tan θ + cot θ = 1.
Show that `(cos^2(45^circ + θ) + cos^2(45^circ - θ))/(tan(60^circ + θ) tan(30^circ - θ)) = 1`
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
