Question

Prove the following cosecA/cosecA-1+cosecA/cosecA+1=2+2tanAtanA?

Answer 1

See below

Explanation:

`cscA/(cscA-1)+cscA/(cscA+1)=`

`(cscA(cscA+1))/((cscA-1)(cscA+1))+(cscA(cscA-1))/((cscA-1)(cscA+1))=`

`(csc^2A+cscA)/(csc^2A-1)+(csc^2A-cscA)/ (csc^2A-1)=`

Apply modified pythagorean identity: `csc^2A-1= cot^2A`

`(2csc^2A)/Cot^2A=`

`2csc^2A*tan^2A=`

`2/cancel(sin^2A)*cancel(sin^2A)/cos^2A=`

`2/cos^2A=`

`2sec^2A=`

Apply pythagorean identity: `1+tan^2A= sec^2A`

`2(1+tan^2A)=`

`2+2tan^2A=`

`2+2tanAtanA quad sqrt`