Question

cos -sin+1/cos+sin-1=cosec+cot​

Answer 1

Step-by-step explanation:

cos-sin+1/cos+sin-1

decide by sin in each term

cot-1+cosec/cot+1-cosec

=cosec+cot-(cosec²-cot²)/cot-cosec+1

=(cosec+cot)-(1-cosec+cot)/(cot-cosec+1)

=cosec+cot

Answer 2

Step-by-step explanation:

cosA-sinA+1/cosA+sinA-1

=(cosA-sinA+1)(cosA+sinA+1)/(cosA+sinA-1)(cosA+sinA+1)

=(cos²A-cosAsinA+cosA+cosAsinA-sin²A+sinA+cosA-sinA+1)/{(cosA+sinA)²-(1)²}

=(cos²A-sin²A+2cosA+1)/(cos²A+2cosAsinA+sin²A-1)

={cos²A+2cosA+(1-sin²A)}/(1+2cosAsinA-1) [∵, sin²A+cos²A=1]

=(cos²A+2cosA+cos²A)/2cosAsinA

=(2cos²A+2cosA)/2cosAsinA

=2cosA(cosA+1)/2cosAsinA

=(cosA+1)/sinA

=cosA/sinA+1/sinA

=cotA+cosecA

=cosecA+cotA (Proved)