# prove that cotA + cosecA -1/ cotA - CosecA + 1 = 1 + cosA - sinA

• 1

cotA + cosecA -1/cotA - cosecA  + 1= cosA/sinA + 1/sinA - sinA/cosA -1/sinA + 1

= (cos2A - sin2A + sinAcosA)/sinAcosA

= (cosA(1 + sinA))/sinAcosA

sorry hv to go solve it completely later hope this musch will help u a bit

• 1

i didnt understand the second and third step...can u plz explain it...

• -4

Alternative method:

cot A + cosec A - 1 /  cot A - cosec A + 1   =  1 + cos A / sin A

LHS : cot A + cosec A - 1 /  cot A - cosec A + 1
[ (cosA/sinA) +(1/sinA) – 1] / [ (cosA/sinA) -(1/sinA) + 1]
{ [cosA + 1-sinA ]/sinA} / { [cosA – 1+sinA ]/sinA }
{ [cosA + 1-sinA ]/sinA} * sinA/[cosA – 1+sinA ]
[cosA + 1-sinA ]/ [cosA – 1+sinA ]
(cosA- sinA) +1/ (cosA+sinA) – 1
{ (cosA- sinA) +1/ (cosA+sinA) – 1}* {(cosA+sinA) +1 /(cosA+sinA) +1}
( Rationalising it by"(cosA+sinA) +1")
{(cosA- sinA)(cosA+sinA) +(cosA- sinA) +(cosA- sinA) +1} /[(cosA+sinA) – 1][(cosA+sinA) +1]

{(cos2A- sin2A)+2cosA+sinA- sinA +1}/ (cosA+sinA)2  -1
{ cos2A-(1- cos2A) + 2cosA+1} / (cos2A+sin2A+2sinA cosA) -1
{ cos2A-1+ cos2A + 2cosA+1}  /  2sinA cosA +11
2cos2A + 2cosA  /  2sinA cosA
2 cosA (cosA +1) / 2cosA (sinA)
1 + cos A / sin A  LHS = RHS  (Hence proved)
IF SATISFIED DO GIVE A THUMB UP
• 61

how did cosA-SinA came 3 times after rationalising??

• -4

shriyansh1994 thats during the time of rationalising ,The time of multiplication it seems its 3 times ,we are simplifying it shriyansh1994 observe carefully.One student had asked the same doubt what you asked but I gave the same solution.He was very satisfied with my same solution which I gave to you and he gave me a thumb up .After observing if you are satisfied do give a thumb up and if any more confusion just mention .

• -2

Gud imad.... thumbs up fr u....

• -7

• -5

NITIN the answer I have given is absolutely correct .But in the question given by shriyansh1994 the RHS is not 1 + cosA - sinA but it must be 1 + cos A / sin A .Do observe my solution carefully and do give you suggestion then,IF YOU ARE SATISFIED DO GIVE A THUMB UP.

• -3

yes he is right.....i verified the queston it is 1+cosA/sinA

• -4

{(cosA- sinA)(cosA+sinA) +(cosA- sinA) +(cosA- sinA) +1} /[(cosA+sinA) – 1][(cosA+sinA) +1]

but how you got  the numerator {(cosA- sinA)(cosA+sinA) +(cosA- sinA) +(cosA- sinA) +1}                                                   is it like this know?? {(cosA- sinA)(cosA+sinA) +(cosA- sinA) +(cosA+ sinA) +1}   ?????

• 3

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

[cotA + cosecA + (cot2A - cosec2A)] / (cotA – cosecA + 1)                        {because  cot2A – cosec2A = -1}

(cotA + cosecA ) + [ (cotA + cosecA) (cotA – cosecA ) ] / (cotA - cosecA + 1)

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

(cotA + cosecA)

cosA / sinA + 1 / sinA

1 + cosA / sinA

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