Advertisements
Advertisements
Question
If `(cot theta ) = m and ( sec theta - cos theta) = n " prove that " (m^2 n)(2/3) - (mn^2)(2/3)=1`
Advertisements
Solution
We have `(cot theta + tan theta ) = m and ( sec theta - cos theta )=n`
Now, `m^2 n = [(cot theta + tan theta )^2 (sec theta - cos theta )]`
=`[(1/tan theta + tan theta )^2 (1/cos theta- cos theta )]`
=`(1+tan^2 theta)^2/tan^2 theta xx ((1-cos^2 theta))/costheta`
=`sec^4 theta/tan^2 theta xx sin^2 theta/ cos theta`
=`sec ^4 theta /(sin^2 theta/cos^2 theta) xx sin^2 theta / cos theta`
=`(cos^2 xxsec^4 theta)/costheta`
=`cos theta sec^4 theta`
=`1/ sec theta xx sec ^4 theta = sec^3 theta`
∴`(m^2 n)^(2/3) =(sec^3 theta )^(2/3) = sec^2 theta`
Again , `mn^2 = [(cot theta + tan theta )( sec theta - cos theta )^2 ]`
=`[(1/tan theta + tan theta).(1/ cos theta - cos theta)^2]`
=`((1+ tan^2 theta))/tan theta xx ((1- cos^2 theta)^2)/cos^2 theta `
=`sec^2 theta/tan theta xx sin^4 theta/cos^2 theta`
=`sec^2 theta/(sintheta/costheta) xx sin^4 theta/ cos^2 theta`
=`(sec^2 xx sin^3 theta)/cos theta`
=`1/ cos^2 theta xx sec^3 theta/ cos theta = tan^3 theta `
∴ `(mn^2)^(2/3) = (tan ^3 theta )^(2/3) = tan^2 theta`
Now ,` (m^2n)^(2/3) - (mn^2)^(2/3)`
=`sec^2 theta - tan^2 theta =1 `
=RHS
Hence proved.
APPEARS IN
RELATED QUESTIONS
If sinθ + cosθ = p and secθ + cosecθ = q, show that q(p2 – 1) = 2p
Prove that `(tan^2 theta)/(sec theta - 1)^2 = (1 + cos theta)/(1 - cos theta)`
Prove the following trigonometric identities.
`(tan^2 A)/(1 + tan^2 A) + (cot^2 A)/(1 + cot^2 A) = 1`
Prove the following trigonometric identities.
`(tan^3 theta)/(1 + tan^2 theta) + (cot^3 theta)/(1 + cot^2 theta) = sec theta cosec theta - 2 sin theta cos theta`
Prove the following identities:
`(costhetacottheta)/(1 + sintheta) = cosectheta - 1`
Prove that:
`(sinA - sinB)/(cosA + cosB) + (cosA - cosB)/(sinA + sinB) = 0`
Show that : `sinA/sin(90^circ - A) + cosA/cos(90^circ - A) = sec A cosec A`
Write the value of `sin theta cos ( 90° - theta )+ cos theta sin ( 90° - theta )`.
If cos A + cos2 A = 1, then sin2 A + sin4 A =
If x = acosθ , y = bcotθ , prove that `a^2/x^2 - b^2/y^2 = 1.`
For ΔABC , prove that :
`sin((A + B)/2) = cos"C/2`
Prove that `sinA/sin(90^circ - A) + cosA/cos(90^circ - A) = sec(90^circ - A) cosec(90^circ - A)`
Find A if tan 2A = cot (A-24°).
If sin θ (1 + sin2 θ) = cos2 θ, then prove that cos6 θ – 4 cos4 θ + 8 cos2 θ = 4
If cos θ = `24/25`, then sin θ = ?
Prove that `costheta/(1 + sintheta) = (1 - sintheta)/(costheta)`
If sec θ = `41/40`, then find values of sin θ, cot θ, cosec θ
Prove that `(1 + sec "A")/"sec A" = (sin^2"A")/(1 - cos"A")`
tan θ × `sqrt(1 - sin^2 θ)` is equal to:
Eliminate θ if x = r cosθ and y = r sinθ.
