Question

please answer this........
prove that​

Answer 1

Answer:-

(Cot A - CosA) / (Cot A + Cos A) = (Cosec A - 1) / (Cosec A + 1)

L.H.S

[(CosA/SinA - CosA)] / [(CosA/SinA + CosA)]

[(CosA - CosASinA) / SinA] / [ (CosA + CosASinA) / SinA]

[CosA - CosASinA] / [CosA + CosASinA]

[CosA(1 - SinA] / CosA(1+Sin A) ] =

(1 - SinA)/ (1 + SinA)

R.H.S

(Cosec A - 1) / (Cosec A + 1)

= (1/SinA - 1) / ( 1/ SinA + 1)

= [ ( 1 - SinA)/Sin A ] / [ ( 1 + SinA) / SinA ]

= (1 - Sin A)/(1 + SinA)

Alternatively,

L.H.S

(1 - Sin A)/ (1 + Sin A)

Divide each term by Sin A

(1/SinA - SinA/SinA) / (1/SinA + SinA/SinA) =

(Cosec A - 1) / (Cosec A + 1) = R.H.S

Hence, proved !

Answer 2

Answer :-

(CotA - CosA) / (CotA + CosA) = (CosecA - 1) / (CosecA + 1)

LHS

[(CosA /SinA - CosA)] / [(CosA /SinA + CosA)]

[(CosA - CosASinA) / SinA)] / [(CosA + CosASinA) / SinA)]

[CosA - CosASinA] / [CosA + CosASinA]

[CosA (1 - SinA)] / [CosA (1 + SinA)] = (1 - SinA) / (1 + SinA)

RHS

(Cosec A - 1) / (Cosec A + 1) = (1/SinA - 1) / (1/SinA + 1)

= [ (1 - SinA)/SinA ] / [ (1 + SinA)/SinA ]

= (1 - SinA) / (1 + SinA)

Alternatively :-

LHS

(1 - SinA) / (1 + SinA)

Now Devide each term by SinA

(1/SinA - SinA/SinA) / (1/SinA + SinA/SinA) = (CosecA - 1) / (CosecA + 1) = RHS

Hence proved !