Fsc Part 2 Mathematics (Complete Solutions)

Q8
Prove the identities:

(i)     sinh2x = 2sinhxcoshx
(ii)    sech2x = 1 - tanh2x
(iii)   csch2x = coth2x - 1

Solution

(i)     sinh2x = 2sinhxcoshx
R.H.S = 2sinhxcoshx
= 2 (
ex - e-x
2
) (
ex + e-x
2
)
=
2
4
(ex - e-x) (ex + e-x)
=
1
2
(ex - e-x) (ex + e-x)
=
1
2
(ex - e-x) (ex + e-x)
=
(ex - e-x) (ex + e-x)
2
=
ex + x + ex - x - e-x + x - e-x - x
2
=
e2x + e0 - e0 - e-2x
2
=
e2x + 1 - 1 - e-2x
2
=
e2x - e-2x
2
 = sinh2x
Hence proved R.H.S = L.H.S

(ii)    sech2x = 1 - tanh2x
R.H.S = 1 - tanh2x
= 1 - (
ex - e-x
ex + e-x
)2
= 1 -
(ex - e-x)2
(ex + e-x)2
=
(ex + e-x)2 - (ex - e-x)2
(ex + e-x)2
=
e2x + e-2x + 2(ex)(e-x) - [e2x + e-2x - 2(ex)(e-x)]
(ex + e-x)2
=
e2x + e-2x + 2(ex - x) - e2x - e-2x + 2(ex - x)
(ex + e-x)2
=
2(e0) + 2(e0)
(ex + e-x)2
=
2(1) + 2(1)
(ex + e-x)2
=
2 + 2
(ex + e-x)2
=
4
(ex + e-x)2
=
(2)2
(ex + e-x)2
= (
2
ex + e-x
)2
 = sech2x
 Hence proved R.H.S = L.H.S

(iii)   csch2x = coth2x - 1
R.H.S = coth2x - 1
= (
ex + e-x
ex - e-x
)2 - 1
=
(ex + e-x)2
(ex - e-x)2
- 1
=
(ex + e-x)2 - (ex - e-x)2
(ex - e-x)2
=
e2x + e-2x + 2(ex)(e-x) - [e2x + e-2x - 2(ex)(e-x)]
(ex - e-x)2
=
e2x + e-2x + 2(ex - x) - e2x - e-2x + 2(ex - x)
(ex - e-x)2
=
2(e0) + 2(e0)
(ex - e-x)2
=
2(1) + 2(1)
(ex - e-x)2
=
2 + 2
(ex - e-x)2
=
4
(ex - e-x)2
=
(2)2
(ex - e-x)2
= (
2
ex - e-x
)2
 = coth2x
 Hence proved R.H.S = L.H.S

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