tan`theta+cottheta=secthetacsctheta`
2 Answers
jeew-m | College Teacher | (Level 1) Educator Emeritus
In trigonometry we know that;
`tantheta = (sintheta)/(costheta)`
`cottheta = 1/(tantheta) = (costheta)/(sintheta)`
`sectheta = 1/(costheta)`
`csctheta = 1/(sintheta)`
`tantheta+cottheta`
`= tantheta+1/(tantheta)`
`= (tantheta)^2/(tantheta)+1/(tantheta)`
`= (1+tan^2theta)/(tantheta)`
We know that;
`1+tan^2theta = sec^2theta = 1/(cos^2theta)`
`tantheta+cottheta`
`= (1+tan^2theta)/(tantheta)`
`= (1/(cos^2theta))/((sintheta)/(costheta))`
`= (costheta)/((cos^2thetaxxsintheta))`
`= 1/((costhetaxxsintheta))`
`= 1/(costheta)xx1/(sintheta)`
`= secthetaxxcsctheta`
So the answer is proved.
`tantheta+cottheta = secthetaxxcsctheta`
aruv | High School Teacher | (Level 2) Valedictorian
Since
`tan(theta)=(sin(theta))/(cos(theta))`
`therefore`
`cot(theta)=1/(tan(theta))=(cos(theta))/(sin(theta))`
`L.H.S.=tan(theta)+cot(theta)`
`=(sin(theta))/(cos(theta))+(cos(theta))/(sin(theta))`
`=(sin^2(theta)+cos^2(theta))/(cos(theta)sin(theta))`
`since`
`sin^2(theta)+cos^2(theta)=1`
`therefore`
`L.H.S.=1/(sin(theta)cos(theta))=1/(sin(theta))xx1/(cos(theta))`
`=cosec(theta)sec(theta)=sec(theta)cosec(theta)`
=R.H.S.