(ii) f′(x)=limh→0f(x+h)−f(x)h =limh→0cosec(x+h)−cosecxh =limh→01sin(x+h)−1sinxh =limh→0sinx−sin(x+h)sin(x+h)sinxh
Using, sinA−sinB=2cos(A+B)2sin(A−B)2 =limh→02cos(x+x+h)2sin(x−x−h)2hsin(x+h)sinx =limh→02cos(2x+h)2sin(−h)2hsin(x+h)sinx =limh→0cos(2x+h)2sin(x+h)sinxlimh→0−sin(h/2)h2
Second limit has value unity, which is already derived in theory class. =cos(x)(sin(x)sinx)(−1)=−cosecxcotx
(iii) f′(x)=limh→0f(x+h)−f(x)h =limh→0sec(x+h)−secxh =limh→01cos(x+h)−1cosxh =limh→0((cosx−cos(x+h))/(cos(x+h)cosx))/h
Using, cosA−cosB=−2sin(A+B)2sin(A−B)2 =limh→0−2sin(x+x+h)2sin(x−x−h)2hcos(x+h)cosx =limh→0−2sin(2x+h)2sin(−h)2hcos(x+h)cosx =limh→0−sin(2x+h)2cos(x+h)cosxlimh→0−sin(h/2)h2
Second limit has value unity, which is already derived in theory class. =limh→0−sin(2x+h)2cos(x+h)cosxlimh→0−sin(h/2)h2=sin(x)(cos(x)cosx)(−1)=secxtanx
(iv) f′(x)=limh→0f(x+h)−f(x)h =limh→0cot(x+h)−cotxh =limh→0cos(x+h)sin(x+h)−cosxsinxh =limh→0cos(x+h)sinx−sin(x+h)cosxsin(x+h)sinxh
Using, sin(A−B)=sinAcosB−cosAsinB =limh→0sin(x−x−h)hsin(x+h)sinx =limh→0sin(−h)hsin(x+h)sinx =limh→01sin(x+h)sinxlimh→0−sin(h)h
Second limit has value unity, which is already derived in theory class. =1sinxsinx(−1)=−cosec2x