Advertisements
Advertisements
Question
If `a cos^3 theta + 3a cos theta sin^2 theta = m, a sin^3 theta + 3 a cos^2 theta sin theta = n`, prove that `(m + n)^(2/3) + (m - n)^(2/3) = 2a^(2/3)`
Advertisements
Solution
`= (a cos^3 theta + 3a cos theta sin^2 theta + a sin^3 theta + 3a cos^2 theta sin theta)^(3/2) + (a cos^3 theta + 3a cos theta sin^2 theta - a sin^3 theta - 3a cos^2 theta sin theta)^(2/3)`
`= a^(1/3) (cos^3 theta + 3 cos theta sin^2 theta + sin^3 theta + 3 cos^2 theta sin theta)^(2/3) + a^(2/3) (cos^3 theta + 3 cos theta sin^2 theta + sin^3 theta - 3 cos^2 theta sin theta)^(2/3)`
`= a^(1/3) [(cos theta + sin theta)^3]^(2/3) + a^(2/3) (cos theta - sin theta)^3]^(2/3)`
`= a^(2/3) [(cos theta + sin theta)^2] + a^(2/3) (cos theta - sin theta)^2`
`= a^(2/3) [cos^2 theta + sin^2 theta - 2sin theta cos theta]`
`= a^(2/3) [cos^2 theta + sin^2 theta + 2 sin theta cos theta] +_ a^(2/3) [cos^2 theta + sin^2 theta - 2 sin theta cos theta]`
`= a^(2/3) [1 + 2 sin theta cos theta] + a^(2/3)[1 - 2 sin theta cos theta]`
`= a^(2/3) [1 + 2 sin theta cos theta + 1 - 2 sin theta cos theta]`
`= a^(1/3) (1 + 1) = 2a^(2/3)`
R.H.S
APPEARS IN
RELATED QUESTIONS
Prove the following identities:
`(i) cos4^4 A – cos^2 A = sin^4 A – sin^2 A`
`(ii) cot^4 A – 1 = cosec^4 A – 2cosec^2 A`
`(iii) sin^6 A + cos^6 A = 1 – 3sin^2 A cos^2 A.`
Prove the following trigonometric identities
cosec6θ = cot6θ + 3 cot2θ cosec2θ + 1
Prove the following trigonometric identities.
`(1 + tan^2 A) + (1 + 1/tan^2 A) = 1/(sin^2 A - sin^4 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 - z^2/c^2 = 1`
Prove the following identities:
`1/(tan A + cot A) = cos A sin A`
Prove the following identities:
`cosA/(1 + sinA) + tanA = secA`
Prove that:
(tan A + cot A) (cosec A – sin A) (sec A – cos A) = 1
If `( cos theta + sin theta) = sqrt(2) sin theta , " prove that " ( sin theta - cos theta ) = sqrt(2) cos theta`
Write the value of `( 1- sin ^2 theta ) sec^2 theta.`
Prove the following identity :
`(1 + tan^2A) + (1 + 1/tan^2A) = 1/(sin^2A - sin^4A)`
Prove the following identity :
`(cotA + cosecA - 1)/(cotA - cosecA + 1) = (cosA + 1)/sinA`
Prove the following identity :
`(tanθ + sinθ)/(tanθ - sinθ) = (secθ + 1)/(secθ - 1)`
If `x/(a cosθ) = y/(b sinθ) "and" (ax)/cosθ - (by)/sinθ = a^2 - b^2 , "prove that" x^2/a^2 + y^2/b^2 = 1`
Evaluate:
sin2 34° + sin2 56° + 2 tan 18° tan 72° – cot2 30°
If sin θ = `1/2`, then find the value of θ.
Prove that cosec2 (90° - θ) + cot2 (90° - θ) = 1 + 2 tan2 θ.
If tan θ – sin2θ = cos2θ, then show that `sin^2θ = 1/2`.
Prove the following:
`tanA/(1 + sec A) - tanA/(1 - sec A)` = 2cosec A
If cosec θ + cot θ = p, then prove that cos θ = `(p^2 - 1)/(p^2 + 1)`
Show that tan4θ + tan2θ = sec4θ – sec2θ.
