Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities
cosec6θ = cot6θ + 3 cot2θ cosec2θ + 1
Advertisements
उत्तर
We need to prove cosec6θ = cot6θ + 3 cot2θ cosec2θ + 1
Solving the L.H.S, we get
`cosec^6 theta = (cosec^2 theta)^3`
`= (1 + cot^2 theta)^3` .......`(1 + cot^2 theta = cosec^2 theta)`
Further using the identity `(a + b)^3 = a^3 + b^3 + 3a^2b + 3ab^2` we get
`(1 + cot^2 theta)^3 = 1 + cot^6 theta + 3(1)^2 (cot^2 theta) + 3(1) (cot^2 theta)^2`
`= 1 + cot^6 theta + 3 cot^2 theta + 3 cot^4 theta`
`= 1 + cot^6 theta + 3 cot^2 theta (1 + cot^2 theta)`
`= 1 + cot^6 theta + 3 cot^2 theta cosec^2 theta` `(using 1 + cot^2 theta = cosec^2 theta)`
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove the following identities:
`1 - cos^2A/(1 + sinA) = sinA`
Prove that:
`(cos^3A + sin^3A)/(cosA + sinA) + (cos^3A - sin^3A)/(cosA - sinA) = 2`
If x cos A + y sin A = m and x sin A – y cos A = n, then prove that : x2 + y2 = m2 + n2
Show that : `sinA/sin(90^circ - A) + cosA/cos(90^circ - A) = sec A cosec A`
Prove the following identities:
`sqrt((1 - cosA)/(1 + cosA)) = sinA/(1 + cosA)`
Prove the following identities:
`(1 + (secA - tanA)^2)/(cosecA(secA - tanA)) = 2tanA`
`sec theta (1- sin theta )( sec theta + tan theta )=1`
`sqrt((1-cos theta)/(1+cos theta)) = (cosec theta - cot theta)`
`sin theta/((cot theta + cosec theta)) - sin theta /( (cot theta - cosec theta)) =2`
If `sqrt(3) sin theta = cos theta and theta ` is an acute angle, find the value of θ .
Prove that secθ + tanθ =`(costheta)/(1-sintheta)`.
Prove the following identity :
`(cotA + cosecA - 1)/(cotA - cosecA + 1) = (cosA + 1)/sinA`
Prove the following identity :
`((1 + tan^2A)cotA)/(cosec^2A) = tanA`
Prove the following identities:
`(sec"A"-1)/(sec"A"+1)=(sin"A"/(1+cos"A"))^2`
Prove that sin4θ - cos4θ = sin2θ - cos2θ
= 2sin2θ - 1
= 1 - 2 cos2θ
If x = a sec θ + b tan θ and y = a tan θ + b sec θ prove that x2 - y2 = a2 - b2.
Prove that `( 1 + sin θ)/(1 - sin θ) = 1 + 2 tan θ/cos θ + 2 tan^2 θ` .
Without using a trigonometric table, prove that
`(cos 70°)/(sin 20°) + (cos 59°)/(sin 31°) - 8sin^2 30° = 0`.
If tan θ = `x/y`, then cos θ is equal to ______.
