Advertisements
Advertisements
प्रश्न
If `(cot theta ) = m and ( sec theta - cos theta) = n " prove that " (m^2 n)(2/3) - (mn^2)(2/3)=1`
Advertisements
उत्तर
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
संबंधित प्रश्न
Prove the following trigonometric identities:
(i) (1 – sin2θ) sec2θ = 1
(ii) cos2θ (1 + tan2θ) = 1
Prove the following trigonometric identities.
(1 + tan2θ) (1 − sinθ) (1 + sinθ) = 1
Prove the following trigonometric identities.
`(1 + cot A + tan A)(sin A - cos A) = sec A/(cosec^2 A) - (cosec A)/sec^2 A = sin A tan A - cos A cot A`
Prove the following identities:
`1/(tan A + cot A) = cos A sin A`
Prove the following identities:
`cot^2A((secA - 1)/(1 + sinA)) + sec^2A((sinA - 1)/(1 + secA)) = 0`
If `tan theta = 1/sqrt(5), "write the value of" (( cosec^2 theta - sec^2 theta))/(( cosec^2 theta - sec^2 theta))`.
Four alternative answers for the following question are given. Choose the correct alternative and write its alphabet:
sin θ × cosec θ = ______
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 a cos θ − b sin θ = c, then a sin θ + b cos θ =
Prove the following identity:
`cosA/(1 + sinA) = secA - tanA`
Prove the following identity :
`sqrt((1 + sinq)/(1 - sinq)) + sqrt((1- sinq)/(1 + sinq))` = 2secq
Proved that cosec2(90° - θ) - tan2 θ = cos2(90° - θ) + cos2 θ.
Prove that: sin4 θ + cos4θ = 1 - 2sin2θ cos2 θ.
Prove that `(tan θ + sin θ)/(tan θ - sin θ) = (sec θ + 1)/(sec θ - 1)`
Prove the following identities.
cot θ + tan θ = sec θ cosec θ
Choose the correct alternative:
sec 60° = ?
Choose the correct alternative:
sec2θ – tan2θ =?
Prove that `(cos^2theta)/(sintheta) + sintheta` = cosec θ
Prove that sec2θ – cos2θ = tan2θ + sin2θ
Prove the following:
`1 + (cot^2 alpha)/(1 + "cosec" alpha)` = cosec α
