Advertisements
Advertisements
Question
`tan theta /((1 - cot theta )) + cot theta /((1 - tan theta)) = (1+ sec theta cosec theta)`
Advertisements
Solution
LHS= `tan theta/((1-cot theta))+ cot theta/((1-tan theta))`
=`tan theta/((1-cos theta/sin theta)) + cot theta/((1-sin theta/cos theta))`
=`(sin theta tan theta)/((sin theta- cos theta))+(cos theta cot theta)/((cos theta - sin theta))`
=`(sin theta xx (sin theta) / (cos theta) cos theta xx (cos theta) / (sin theta))/((sin theta - cos theta))`
=`((sin ^2 theta cos ^2 theta)/(cos theta sin theta))/((sin theta-cos theta))`
=`( sin ^3 theta - cos ^3 theta)/(cos theta sin theta (sin theta - cos theta))`
=` ((sin theta - cos theta)(sin ^2 theta + sin theta cos theta + cos ^2theta ))/(cos theta sin theta (sin theta- costheta))`
=`(1+ sin theta cos theta)/(cos theta sin theta)`
=`1/(cos theta sin theta)+(sin theta cos theta)/(cos theta sin theta)`
=`1/(cos theta sin theta)+ (sin theta cos theta)/(cos theta sin theta)`
=`sectheta cosec theta +1`
=`1+ sec theta cosec theta`
=RHS
APPEARS IN
RELATED QUESTIONS
(secA + tanA) (1 − sinA) = ______.
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`(sin theta-2sin^3theta)/(2cos^3theta -costheta) = tan theta`
Prove that
`sqrt((1 + sin θ)/(1 - sin θ)) + sqrt((1 - sin θ)/(1 + sin θ)) = 2 sec θ`
Prove the following identities:
cot2 A – cos2 A = cos2 A . cot2 A
Prove that:
`tanA/(1 - cotA) + cotA/(1 - tanA) = secA "cosec" A + 1`
Show that : `sinAcosA - (sinAcos(90^circ - A)cosA)/sec(90^circ - A) - (cosAsin(90^circ - A)sinA)/(cosec(90^circ - A)) = 0`
Prove the following identities:
`((cosecA - cotA)^2 + 1)/(secA(cosecA - cotA)) = 2cotA`
`cot^2 theta - 1/(sin^2 theta ) = -1`a
Write the value of `(1 - cos^2 theta ) cosec^2 theta`.
If `secθ = 25/7 ` then find tanθ.
What is the value of 9cot2 θ − 9cosec2 θ?
Prove the following identity :
secA(1 + sinA)(secA - tanA) = 1
Prove the following identity:
`cosA/(1 + sinA) = secA - tanA`
Prove the following identity :
`sin^2Acos^2B - cos^2Asin^2B = sin^2A - sin^2B`
Prove the following identity :
`cos^4A - sin^4A = 2cos^2A - 1`
Prove that `sqrt(2 + tan^2 θ + cot^2 θ) = tan θ + cot θ`.
Prove that: `(sec θ - tan θ)/(sec θ + tan θ ) = 1 - 2 sec θ.tan θ + 2 tan^2θ`
If 5 sec θ – 12 cosec θ = 0, then find values of sin θ, sec θ
Prove that `(sintheta + "cosec" theta)/sin theta` = 2 + cot2θ
