$\"\"$

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(a).

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Prove that $\"\"$ .

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Consider $\"\"$ then $\"\"$ .

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The equation is $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$ .

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$\"\"$

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$\"\"$ .

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Definition of hyperbolic identities : $\"\"$ .

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$\"\"$

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$\"\"$

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Substitue $\"\"$ in the above equation

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$\"\"$ .

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Susbtitute $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$ .

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$\"\"$

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(b).

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Prove that $\"\"$ .

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Consider $\"\"$ then $\"\"$ .

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The equation is $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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Substitue $\"\"$ and $\"\"$ in the above equation

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$\"\"$

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$\"\"$

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(c).

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Prove that $\"\"$ .

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Consider $\"\"$ then $\"\"$ .

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The equation is $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitue $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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Therefore $\"\"$ .

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Substitue $\"\"$ in the above equation

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$\"\"$ .

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$\"\"$

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(d).

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Prove that $\"\"$

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Consider $\"\"$ then $\"\"$ .

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The equation is $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Susbtitue $\"\"$ in the above equation

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$\"\"$

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Substitue $\"\"$ in the above quation

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$\"\"$ .

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$\"\"$

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(e).

\

Prove that $\"\"$ .

\

Consider $\"\"$ then $\"\"$ .

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The equation is $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$ .

\

$\"\"$

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$\"\"$ .

\

$\"\"$

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Substitue $\"\"$ and $\"\"$ in the above equation

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$\"\"$ .

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$\"\"$

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(a). $\"\"$ .

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(b). $\"\"$ .

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(c). $\"\"$ .

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(d). $\"\"$ .

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(e). $\"\"$ .