Q8. Prove ( COSA-SİNA+1) / (COSA+sinA-1) = cosecA + cotA.
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Step-by-step explanation:
Q8. Prove ( COSA-SİNA+1) / (COSA+sinA-1) = cosecA + cotA.
⇒ Given ⇔ ( COSA-SİNA+1) / (COSA+sinA-1) = cosecA + cotA.
LHS = ( cosA-sinA+1) / (cosA+sinA-1) --- [multiply this equation by (1/sinA)]
= [(1/sinA)*[(cosA-sinA+1)] / [(1/sinA)*(cosA+sinA-1)]
= [(cosA/sinA)-(sinA/sinA)+(1/sinA)] / [(cosA/sinA)+(sinA/sinA)-(1/sinA)]
= [cotA - 1 + cosecA] / [cotA + 1 - csecA]
-------------[ value⇔(cosA/sinA)=cotA,(sinA/sinA)=1,(1/sinA)=cosecA]
= [cotA + cosecA - 1] / [1 + cotA - cosecA]
= [cotA + cosecA + (cot²A - cosec²A)] / [1 + cotA - cosecA]
-------------[ we know 1 + cot²A = cosec²A ∴ -1 = cot²A - cosec²A]
= [cotA+cosecA+(cotA +cosecA)*(cotA - cosecA)]/[1 + cotA - cosecA]
-------------[ we know a² - b² = (a + b)*(a - b)]
= [(cotA + cosecA)*(1 + cotA - cosecA)] / [1 + cotA - cosecA]
-------------{we know [a+(a*b)] = a*(1+b)}
= [(cotA + cosecA)/1] * [(1 + cotA - cosecA)/(1 + cotA - cosecA)]
= (cotA + cosecA)
LHS = RHS
∴ it is proved
( cosA - sinA + 1) / (cosA + sinA - 1) = cosecA + cotA.