Question

prove that 1 by cosec theta minus cot theta minus 1 by sin theta equals to 1 by sin theta minus 1 by cosec theta + cot theta​

Answer 1

Answer:

see attachment

Step-by-step explanation:

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Answer 2

✧ANSWER✧

$\:\:\:\:$

$\bf\footnotesize\dfrac{1}{(cosፀ - cotፀ)}$ $\sf -$ $\bf\footnotesize\dfrac{1}{(sinፀ)}$ $\sf =$ $\bf\footnotesize\dfrac{1}{(sinፀ)}$ $\sf -$ $\bf\footnotesize\dfrac{1}{(cosecፀ + cotፀ)}$

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$\:\:\:\:\:$$\sf LHS$ $\sf =$ $\bf\small\dfrac{1}{(cosecፀ - cotፀ)}$ $\sf =$ $\bf\small\dfrac{1}{(sinፀ)}$

$\:\:$

$\implies$ $\bf\small\dfrac{1}{(cosecፀ - cotፀ)}$ $\sf ×$ $\bf\small\dfrac{(cosecፀ + cotፀ)}{(cosecፀ + cotፀ)}$ $\sf -$ $\sf cosecፀ$

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$\:\:\:\:$$\implies$ $\bf\dfrac{(cosecፀ + cotፀ)}{(cosec²ፀ + cot²ፀ)}$ $\sf -$ $\sf cosecፀ$

$\:\:$

$\:\:\:\:$$\implies$ $\sf (cosecፀ + cotፀ)-cosecፀ$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$\underline{\boxed{\sf{=\frak{\red{cotፀ}}}}}$

━━━━━━━━━━━━━━━━━━━━━━━━━

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$\:\:$$\sf RHS$ $\sf =$ $\bf\dfrac{1}{(sinፀ)}$ $\sf -$ $\bf\dfrac{1}{(cosecፀ + cotፀ)}$

$\:\:$

$\:\:$$\implies$ $\sf cosecፀ$ $\sf -$ $\bf\dfrac{1}{(cosecፀ + cotፀ)}$

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$\implies$ $\sf cosecፀ$ $\sf -$ $\bf\dfrac{1}{(cosecፀ + cotፀ)}$ $\sf ×$ $\bf\dfrac{(cosecፀ - cotፀ)}{(cosecፀ - cotፀ)}$

$\:\:$

$\:\:$$\implies$ $\sf cosecፀ$ $\sf -$ $\bf\dfrac{(cosecፀ - cotፀ)}{(cosec²ፀ - cot²ፀ)}$

$\:\:$

$\:\:$$\implies$ $\sf cosecፀ$ $\sf -$ $\sf (cosecፀ-cotፀ)$

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$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$\underline{\boxed{\sf{=\frak{\green{cotፀ}}}}}$

$[/tex][tex]\:\:$

${\bold { \underline{\large{\therefore\:LHS \:=\: RHS}}}}$