if cosecx- cotx= 1/2, then cosx is equal to
Answers
As we know the trigonometric identity,
cosec2x−cot2x=1Or
(cosec x−cotx)(cosec x+cotx)=1
[As we know that
a2−b2=(a−b)(a+b)]
Here it is given that
cosec x −cotx = 12.....(1) , now putting this in above equation we get,
(cosec x−cotx)(cosec x+cotx)=1 1/2(cosec x+cotx)=1
cosec x+cotx=2........(2)now adding equation (1) and (2) we get,
2cosec x = 2+1/2=5/2or
cosec x = 5/2×2=5/4
So sin x = 1
cosec x=45
Now as we know that , cos x = 1−sin2x−−−−−−−−√=1−(4/5)2−−−−−−−√=25−16/25−−−−−√=9/25−−√=3/5[ Here only positive value will be considered because all other trigonometric ratios are in first quadrant]
From identity cosec²x - cot²x =1 ⇒ [ cosec(x) + cot(x) ] [cosec(x) - cot(x)] = 1 ⇒ cosec(x) + cot(x) = 2 -------- (2) ; cosec(x) - cot (x) = 1/2 -----------(1)
By solving (1) & (2) we get 2cosec(x) = 5/2 ⇒ cosec(x) = 5/4
⇒ sin(x) = 4/5 ⇒ cos(x) = 3/5