Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
sec6θ = tan6θ + 3 tan2θ sec2θ + 1
उत्तर
We need to prove `sec^6 theta = tan^6 theta + 3 tan^2 theta sec^2 theta + 1`
Solving the L.H.S, we get
`sec^6 theta = (sec^2 theta)^3`
`= (1 + tan^2 theta)^3`
Further using the identity `(a + b)^3 = a^3 + b^3 + 3a^2b + 3ab^2` , we get
`(1 + tan^2 theta)^3 = 1 + tan^6 theta + 3(1)^2 (tan^2 theta) + 3(1)(tan^2 theta)^2`
`= 1 + tan^6 theta + 3 tan^2 theta + 3 tan^4 theta`
`= 1 + tan^6 theta + 3 tan^2 theta + 3 tan^4 theta`
`= 1 + tan^6 theta + 3 tan^2 theta (1 + tan^2 theta)`
`= 1 + tan^6 theta + 3 tan^2 theta sec^2 theta` (using `1 + tan^2 theta = sec^2 theta`)
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove the following trigonometric identities.
`cot theta - tan theta = (2 cos^2 theta - 1)/(sin theta cos theta)`
Prove the following identities:
`cosA/(1 + sinA) + tanA = secA`
Prove the following identities:
cosec4 A (1 – cos4 A) – 2 cot2 A = 1
`(1-cos^2theta) sec^2 theta = tan^2 theta`
`cot^2 theta - 1/(sin^2 theta ) = -1`a
If tan A = n tan B and sin A = m sin B , prove that `cos^2 A = ((m^2-1))/((n^2 - 1))`
Four alternative answers for the following question are given. Choose the correct alternative and write its alphabet:
sin θ × cosec θ = ______
Prove the following identity :
`(secA - 1)/(secA + 1) = sin^2A/(1 + cosA)^2`
Find the value of x , if `cosx = cos60^circ cos30^circ - sin60^circ sin30^circ`
A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60° to 45° in 2 minutes. Find the speed of the boat in m/min.
Prove that `(sin (90° - θ))/cos θ + (tan (90° - θ))/cot θ + (cosec (90° - θ))/sec θ = 3`.
Prove that `sqrt((1 + sin θ)/(1 - sin θ))` = sec θ + tan θ.
Without using a trigonometric table, prove that
`(cos 70°)/(sin 20°) + (cos 59°)/(sin 31°) - 8sin^2 30° = 0`.
Prove the following identities.
`sqrt((1 + sin theta)/(1 - sin theta)) + sqrt((1 - sin theta)/(1 + sin theta))` = 2 sec θ
If tan θ = `7/24`, then to find value of cos θ complete the activity given below.
Activity:
sec2θ = 1 + `square` ......[Fundamental tri. identity]
sec2θ = 1 + `square^2`
sec2θ = 1 + `square/576`
sec2θ = `square/576`
sec θ = `square`
cos θ = `square` .......`[cos theta = 1/sectheta]`
Prove that cot2θ – tan2θ = cosec2θ – sec2θ
If 1 + sin2α = 3 sinα cosα, then values of cot α are ______.
If 5 tan β = 4, then `(5 sin β - 2 cos β)/(5 sin β + 2 cos β)` = ______.
Prove the following that:
`tan^3θ/(1 + tan^2θ) + cot^3θ/(1 + cot^2θ)` = secθ cosecθ – 2 sinθ cosθ
`1/sin^2θ - 1/cos^2θ - 1/tan^2θ - 1/cot^2θ - 1/sec^2θ - 1/("cosec"^2θ) = -3`, then find the value of θ.
