Sol. Let 
Required
principal value 
E2. 
Sol. Let cos?1 
where
Required
principal value 
E3. cosec?1 (2)
Sol. Let cosec?1 (2) = y,
Where 
Required
principal value 
E4. 
Sol. Let 
Where 
Required
principal value 
E5. 
Sol. Let y =
cos?1 
where 
Required
principal value 
.
E6. tan?1(?1)
Sol. Let tan?1
= y, where
tan
y = ? 1
tan
y = 
Required
principal value 
E7. 
Sol. Let sec?1
Where 
Required
principal value 
E8. 
Sol. Let
where
Required
principal value
E9. 
Sol. 
where
Required
principal value 
E10. 
Sol. Let 
Required
principal value
Find all the
values of the following :
E11. 
Sol. We know
that
Also, 
E12. 
Sol. We known
that 
and 
E13. If sin?1x
= y, then
(A) 
(B)
(C) 0< y <
(D)
Sol. We know
that the range of the principal value branch of sin?1 is 
Ans
(B) is correct.
E14. 
is
equal to
(A) (B)
(C) 
(D)
Sol. We know
that
and 
NCERT TEXT
BOOKS EXERCISE 2.2
Prove the following :
E1. 
Sol. Let sin?1x
= 
L.H.S.
= 3 sin?1 x = 
,,,
(1)
= R.H.S. =
sin?1 (3x ? 4x3)
= sin?1
(
)
= 
?
(2)
from
(1) and (2), 3 sin?1x(4x3 ? 3x).
E2. 3 cos?1x
= cos?1(4x3 ? 3x), 
.
Sol. 
L.H.S.
= 3 sin?1 x = ,,,
(1)
= R.H.S. = cos?1
(4x3 ? 3x)
= cos?1
(cos 3 
)
= ..
(2)
from
(1) and (2) 3 cos?1x = cos?1 (4x3 ? 3x),
E3. 
Sol. L.H.S. 
Hence proved.
E4. 
Sol. L.H.S. = 2
tan?1 
Hence proved.
Write the
following functions in the simplest form
E5. 
Sol. Let x =
tanq
E6. 
Sol. Let x =
cosec 
where is
required simplest form.
E7. 
Sol. We write
it is
which is
required simplest form.
E8. 
Sol. We write
it as
which is
required simplest form.
E9.
Answer:
We know that
and 
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