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  3. 1-prove the identity. cosh(x) + sinh(x) = ex co

1-Prove the identity. cosh(x) + sinh(x) = ex cosh(x) + sinh(x) = 1 2 ex + e?x + 1 2    = 1 2    = 2-Prove the identity. sinh(x + y)...

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Question

Basic Math

1-Prove the identity.

cosh(x) + sinh(x) = ex

cosh(x) + sinh(x)=
1
2
ex + e?x
+
1
2
  
=
1
2
  
=

2-Prove the identity.

sinh(x + y) = sinh(x) cosh(y) + cosh(x) sinh(y)

sinh(x) cosh(y) + cosh(x) sinh(y)=
1
2
(ex ? e?x)
1
2
  
+
1
2
(ex + e?x)
1
2
  
=
1
4
  + ex ? y ? e?x + y ? e?x ?y
+
ex + y ? ex ? y + e?x + y ?e?x ? y
=
1
4
  ? 2e?x ? y
=
1
2
ex + y ? e?(x + y)
=

3-Prove the identity.

sinh(2x) = 2 sinh(x) cosh(x)

sinh(2x)=sinh
x +   
=sinh(x)   + cosh(x) sinh(x)
=

4-Prove the identity.

1 + tanh(x)
1 ? tanh(x)

= e2x

1 + tanh(x)
1 ? tanh(x)
=
1 +
cosh(x)
1 ?
cosh(x)
=
cosh(x) +
cosh(x) ?
=
1
2
ex + e?x
+
1
2
(ex ? e?x
1
2
ex + e?x
?
1
2
(ex ? e?x
=
e?x
=

5-If

tanh(x) =

4
5

,

find the values of the other hyperbolic functions atx.

sinh(x)=
cosh(x)=
coth(x)=
sech(x)=
csch(x)=

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