Hello friends now we see the derivative
of trigonometric function with proof the
topic number 6 is FX is equal to cosec
X now let's see the proof first of all
we start with F of X is equal to cosec X
therefore the next step is F of X plus h
is equal to cosec in bracket X plus h
then by first principle of derivative
f dash X is equal to limit H tends to 0
f of X plus h minus FX whole thing
divided by H so is equal to limit H
tends to 0 now for f of X plus h we
substitute for say x plus h minus for F
X we substitute Co psychics and then
whole thing divided by H so is equal to
limit H tends to 0 now we apply Twingo
formula that cosec theta is equal to 1
upon sine theta and therefore here for
cosec x plus h we write 1 upon sine in
bracket x plus h then minus 4 passe
cakes we write 1 upon sine x and then
whole thing divided by h the next step
will be here we go for cross multiplying
so that gives us limit H tends to 0 sine
X minus sine of X plus h upon sine of X
plus h in 2 sine X into H
now here we apply formula that sine c
minus sine D that equal to 2 cos bracket
C plus D upon 2 into sine bracket C
minus D upon 2 in present case C
represent X and D represent X plus h so
the next step will be is equal to limit
H tends to 0 to cause in bracket X plus
X + H upon 2 into sine in bracket X
minus X minus H upon 2 and then whole
thing divided by sine of X plus h in 2
sine X into H now we are 2 is a constant
term so we take outside the limit so the
next step will be 2 into maybe H tends
to 0
cause in bracket X plus X 2 X + H upon 2
into now here in bracket plus 6 minus 6
we cancel so we left with sine of minus
H by 2 so that can be written as minus
limit H tends to 0 sine of H by 2 upon H
by 2 into 1 by 2 then whole thing
divided by limit H tends to 0 sine of X
plus h into sine X therefore the next
step will be is equal to 2 into limit H
tends to 0 cos of 2 X + H upon 2 rewrite
as it is then into now this minus sign
we take initially so we have minus 2
into labeed H tends to 0 sine of H by 2
whole thing divided by H by 2 but I
can't complete into 1 by 2 divided by
the limit H tends to 0 sine of X plus h
into sine X
is equal to minus two into now here for
H we substitute zero so we have cos 2x
plus zero upon 2 into now sine H by 2
upon H by 2 as a limit H tends to 0
gives us 1 into 1 by 2 and then whole
thing divided by sine in bracket X plus
0 into sine X so is equal to minus 2
into in bracket 2 to cancel from
numerator denominator so we left with
only cos x into 1 by 2 and in
denominator we have sine X into sine X
now you will be canceled two from
numerator denominator and the remaining
terms can be written as minus 1 upon
sine X into cos x upon sine X and that
gives minus cosec X into now cos upon
sine News cortex so in this way we prove
the derivative of cosecant is minus
cosec X into court X so this is required
proof for the function FX is equal to
cosec X thank you
