Advertisements
Advertisements
Question
If cot θ + tan θ = x and sec θ – cos θ = y, then prove that `(x^2y)^(2/3) – (xy^2)^(2/3)` = 1
Advertisements
Solution
Given cot θ + tan θ = x and sec θ – cos θ = y
x = cot θ + tan θ
x = `1/tan theta + tan theta`
= `(1 + tan^2 theta)/tan theta`
= `(sec^2 theta)/tan theta`
= `(1/cos^2theta)/(sin theta/costheta`
= `1/(cos theta sin theta)`
y = sec θ – cos θ
= `1/cos theta - cos theta`
= `(1 - cos^2 theta)/cos theta`
y = `(sin^2 theta)/costheta`
= `[1/(cos^2thetasin^2theta) xx (sin^2theta)/costheta]^(2/3) - [1/(cos theta sin theta) xx (sin^4 theta)/(cos^2 theta)]^(2/3)`
= `[1/(cos^3theta)]^(2/3) - [(sin^3 theta)/(cos^3 theta)]^(2/3)`
= `[1/(cos^2 theta)] - [(sin^2 theta)/(cos^2 theta)]`
= `[(1 - sin^2 theta)/(cos^2 theta)]`
= `[(cos^2 theta)/(cos^2 theta)]`
= 1
L.H.S = R.H.S
⇒ `(x^2y)^(2/3) – (xy^2)^(2/3)` = 1
APPEARS IN
RELATED QUESTIONS
`"If "\frac{\cos \alpha }{\cos \beta }=m\text{ and }\frac{\cos \alpha }{\sin \beta }=n " show that " (m^2 + n^2 ) cos^2 β = n^2`
Prove the following trigonometric identities
`cos theta/(1 - sin theta) = (1 + sin theta)/cos theta`
Prove the following trigonometric identities
tan2 A + cot2 A = sec2 A cosec2 A − 2
Prove the following identities:
(sec A – cos A) (sec A + cos A) = sin2 A + tan2 A
If `cosA/cosB = m` and `cosA/sinB = n`, show that : (m2 + n2) cos2 B = n2.
What is the value of (1 + tan2 θ) (1 − sin θ) (1 + sin θ)?
Prove the following identity :
cosecθ(1 + cosθ)(cosecθ - cotθ) = 1
If x sin3θ + y cos3 θ = sin θ cos θ and x sin θ = y cos θ , then show that x2 + y2 = 1.
If tan θ = `13/12`, then cot θ = ?
Prove that
`(cot "A" + "cosec A" - 1)/(cot"A" - "cosec A" + 1) = (1 + cos "A")/"sin A"`
