$\"\"$

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(a)

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Show that $\"\"$ .

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Here $\"\"$ , $\"\"$ .

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$\"\"$ can be written as $\"\"$ .

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$\"\"$

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Apply reciprocal identity : $\"\"$ .

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$\"\"$

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Apply Pythagorean identity : $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in above equation.

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$\"\"$ .

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Thus, $\"\"$ .

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$\"\"$

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Show that $\"\"$ .

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$\"\"$ can be written as $\"\"$ .

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$\"\"$

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Apply quotient iidentity : $\"\"$ .

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Apply reciprocal identity : $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in above equation.

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$\"\"$ .

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Thus, $\"\"$ .

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$\"\"$

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(b)

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Show that $\"\"$ .

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Here $\"\"$ , $\"\"$ .

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Double-angle identity : $\"\"$ .

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Therefore, $\"\"$ can be written as $\"\"$ .

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$\"\"$

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Apply reciprocal identity : $\"\"$ .

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$\"\"$

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Apply Pythagorean identity : $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in above equation.

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$\"\"$

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Thus, $\"\"$ .

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$\"\"$

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Show that $\"\"$ .

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Double-angle identity : $\"\"$ .

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Therefore, $\"\"$ can be written as $\"\"$ .

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$\"\"$

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Apply quotient iidentity : $\"\"$ .

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Apply reciprocal identity : $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in above equation.

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$\"\"$ .

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Thus, $\"\"$ .

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$\"\"$

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(c)

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Show that $\"\"$ .

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From (b), $\"\"$ and $\"\"$ .

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Consider $\"\"$ .

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

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$\"\"$

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From (b) $\"\"$ .

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$\"\"$

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Thus, $\"\"$ .

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$\"\"$

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(a) $\"\"$ and $\"\"$ .

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(b) $\"\"$ and $\"\"$ .

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(c) $\"\"$ .