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
If `x/a=y/b = z/c` show that `x^3/a^3 + y^3/b^3 + z^3/c^3 = (3xyz)/(abc)`.
Prove the following trigonometric identities.
`sin theta/(1 - cos theta) = cosec theta + cot theta`
Prove the following trigonometric identities.
(1 + tan2θ) (1 − sinθ) (1 + sinθ) = 1
Prove the following trigonometric identities.
(sec A + tan A − 1) (sec A − tan A + 1) = 2 tan A
Prove the following identities:
`(1 + sin A)/(1 - sin A) = (cosec A + 1)/(cosec A - 1)`
Prove the following identities:
`1/(sinA + cosA) + 1/(sinA - cosA) = (2sinA)/(1 - 2cos^2A)`
Prove that:
`1/(sinA - cosA) - 1/(sinA + cosA) = (2cosA)/(2sin^2A - 1)`
`(cos^3 θ + sin^3 θ)/(cos θ + sin θ) + (cos ^3 θ - sin^3 θ)/(cos θ - sin θ) = 2`
Find the value of ` ( sin 50°)/(cos 40°)+ (cosec 40°)/(sec 50°) - 4 cos 50° cosec 40 °`
Write True' or False' and justify your answer the following:
\[ \cos \theta = \frac{a^2 + b^2}{2ab}\]where a and b are two distinct numbers such that ab > 0.
If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then
If x = asecθ + btanθ and y = atanθ + bsecθ , prove that `x^2 - y^2 = a^2 - b^2`
Evaluate:
`(tan 65^circ)/(cot 25^circ)`
If sin θ + cos θ = `sqrt(3)`, then prove that tan θ + cot θ = 1.
Prove that `(tan^2 theta - 1)/(tan^2 theta + 1)` = 1 – 2 cos2θ
If a cos θ – b sin θ = c, then prove that (a sin θ + b cos θ) = `± sqrt(a^2 + b^2 - c^2)`
sin2θ + sin2(90 – θ) = ?
If cos θ = `24/25`, then sin θ = ?
If 5 tan β = 4, then `(5 sin β - 2 cos β)/(5 sin β + 2 cos β)` = ______.
(sec2 θ – 1) (cosec2 θ – 1) is equal to ______.
