you should prove them

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you should prove them

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tan² A - Sin² A = Sin²A / Cos² A - Sin² A

= Sin² A sec²A - Sin²A

= Sin ² A ( sec²A - 1)

= sin² A tan²A = sin² A sin²A / cos²A

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ii

multiply the two terms to get

1+tan A+sec A+cot A+1+cotA SecA - CosecA - cosecA tan A - cosecA secA

= 2 + tanA + secA +cosA/sinA + cosA/sinA *1/cosA - 1/sinA - 1/sinA*sinA/cosA - 1/sinA * 1/cosA

= 2 + sinA/cosA + 1/cosA +CosA/sinA+ 1/sinA -1/sinA - 1/cosA - 1/sinA*1/cosA

= 2 + sinA/cosA + cosA/sinA - 1/sinA * 1/cosA

= 2 + (sin²A +cos²A - 1 ) / (sinA cosA)

= 2

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LHS 1/(cosecA- cotA) - 1/sin A

multiply numerator and denominator with (coseA+ cotA)

(cosecA + cotA)/ 1 - cosec A = cot A

RHS 1/sinA - 1/ (cosecA + cotA)

multiply numerator and denominator with (cosec A - cot A)

cosec A - (cosecA - cotA) / (cosec²A - cot²A)

= cot A

LHS = RHS = cot A

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(cot A + tan B) / ( cot B + tan A) =

= (cosA / sinA + sin B / cosB ) / (cosB/sinB + sin A/ cosA)

= (cosA cosB + sinA sin B) * CosA sin B / [ (cosA cosB + sinA sinB) * sinA cos B ]

= cot A tan B

as the summation term cancels.