Question

prove that cos A cosecA-sinA secA/cosA+sinA=cosecA-secA

Answer 1

cos(A)csc(A) - sin(A)sec(A) 
--------------------------------------... 
.....cos(A) + sin(A) 

....cos(A)/sin(A) - sin(A)/cos(A)........sin(A)cos(A) 
=-------------------------------------... x ---------------------- 
..........cos(A) + sin(A).....................sin(A)cos(A) 

.......cos^2(A) - sin^2(A) 
= ----------------------------------------... 
......sin(A)cos^2(A) + cos(A)sin^2(A) 

......cos^2(A) - sin^2(A) 
=------------------------------------- 
...sin(A)cos(A)[cos(A) + sin(A)] 

....[cos(A) + sin(A)][cos(A) - sin(A)] 
=-------------------------------------... cancel cos(A) + sin(A) 
...sin(A)cos(A)[cos(A) + sin(A)] 

....cos(A) - sin(A) 
=--------------------------- 
...sin(A)cos(A) 


= 1/sin(A) - 1/cos(A) 

=csc(A) - sec(A) prove//

Answer 2

cosA×1/sinA-sinA×1/cosA
____________________
cosA+ sinA


= cosA/sinA —sinA/cosA
___________________
cosA+sinA

=cos²A –sin²A
____________
sinA×cosA/cosA+sinA


=cos²A–sin²A
————————
sinA×cosA (cosA+sinA)

= (cosA +sinA)(cosA – sinA)
_______________________
sinA×cosA(cosA+sinA)


= cosA–sinA
___________
sinA×cosA

=cosA/sinA×cosA — sinA/sinA×cosA

=1/sinA – 1/cosA

= cosecA–secA

=RHS PROVED.