Advertisements
Advertisements
Question
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`
Advertisements
Solution
Given:
`x = a cos^3 theta`
`=> x/a = cos^3 theta`
`x = b sin^3 theta`
`=> y/b = sin^3 theta`
We have to prove `(x/a)^(2/3) + (y/b)^(2/3) = 1`
We know that `sin^2 theta + cos^2 theta =1`
So we have
`(x/a)^(2/3) + (yb)^(2/3) = (cos^2 theta)^(2/3) + (sin^3 theta)^(2/3)`
`=> (x/a)^(2/3) + (y/b)^(2/3) = cos^2 theta + sin^2 theta`
`=> (x/a)^(2/3) + (y/b)^(2/3) = 1`
Hence proved
APPEARS IN
RELATED QUESTIONS
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`(1+ secA)/sec A = (sin^2A)/(1-cosA)`
[Hint : Simplify LHS and RHS separately.]
Prove that (1 + cot θ – cosec θ)(1+ tan θ + sec θ) = 2
Prove the following trigonometric identities.
`(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 + sec A))`
If sin A + cos A = m and sec A + cosec A = n, show that : n (m2 – 1) = 2 m
If `(cosec theta - sin theta )= a^3 and (sec theta - cos theta ) = b^3 , " prove that " a^2 b^2 ( a^2+ b^2 ) =1`
If 5 `tan theta = 4,"write the value of" ((cos theta - sintheta))/(( cos theta + sin theta))`
Prove that:
`"tanθ"/("secθ" – 1) = (tanθ + secθ + 1)/(tanθ + secθ - 1)`
Prove the following identity :
`sinA/(1 + cosA) + (1 + cosA)/sinA = 2cosecA`
Prove the following identity :
`(secθ - tanθ)^2 = (1 - sinθ)/(1 + sinθ)`
Prove the following identity :
`sec^4A - sec^2A = sin^2A/cos^4A`
Prove that sin4θ - cos4θ = sin2θ - cos2θ
= 2sin2θ - 1
= 1 - 2 cos2θ
If tan α = n tan β, sin α = m sin β, prove that cos2 α = `(m^2 - 1)/(n^2 - 1)`.
Prove the following identities.
tan4 θ + tan2 θ = sec4 θ – sec2 θ
Choose the correct alternative:
cot θ . tan θ = ?
Prove that cos2θ . (1 + tan2θ) = 1. Complete the activity given below.
Activity:
L.H.S = `square`
= `cos^2theta xx square .....[1 + tan^2theta = square]`
= `(cos theta xx square)^2`
= 12
= 1
= R.H.S
Prove that sec2θ + cosec2θ = sec2θ × cosec2θ
If 3 sin θ = 4 cos θ, then sec θ = ?
Prove that `"cot A"/(1 - tan "A") + "tan A"/(1 - cot"A")` = 1 + tan A + cot A = sec A . cosec A + 1
If 2sin2θ – cos2θ = 2, then find the value of θ.
If sinθ = `11/61`, then find the value of cosθ using the trigonometric identity.
