Let's Start By Expanding The Expression:- csc(u-v).
So, We have,
csc(u - v)
= 1/sin(u - v)
= 1/(sin u cos v - cos u sin v).................(i)
[Expanded using the Identity sin(A - B) = sinAcosB - cosAsinB]
Now, We have to find sin v and cos u too.
We all know,
color(white)(xxx)sin^2theta + cos^2theta = 1
rArr sin^2theta = 1 - cos^2 theta
rArr sin theta = +- sqrt(1 - cos^2 theta)
And, Vice Versa.
So, sin v = +-sqrt(1 - cos^2 v) = +-sqrt(1 - (-3/5)^2) = +-sqrt((25 - 9)/25) = +-4/5
And, cos u = =-sqrt(1 - sin^2 u) = +-sqrt(1 - (5/13)^2) = +-sqrt((169 - 25)/169) = +-12/13
As, sin v and cos u have both positive and negative values, there will be FOUR values (maybe different or same) [as 4 combination 2 = 4] for csc(u - v).
So,
From (i),
First Value for csc(u - v) = 1/((cancel5/13 xx-3/cancel5 )- (4/5 xx 12/13)) = 1/(-3/13 - 48/65) = 1/(-(15 + 48)/65) = -65/63
Second Value for csc(u - v) = 1/((cancel5/13 xx-3/cancel5 )- (4/5 xx -12/13)) = 1/(-3/13 + 48/65) = 1/((48 - 15)/65) = 65/33
Third Value for csc(u - v) = 1/((cancel5/13 xx-3/cancel5 )- (-4/5 xx 12/13)) = 1/(-3/13 - 48/65) = 1/(-(15 + 48)/65) = -65/63
Fourth Value for csc(u - v) = 1/((cancel5/13 xx-3/cancel5 )- (-4/5 xx -12/13)) = 1/(-3/13 + 48/65) = 1/((48 - 15)/65) = 65/33
We can see First and Third and Second and Fourth values for csc(u -v) are same.
So, Basically there are two values for csc (u - v).
In the End,
csc(u - v) = -65/63, 65/33 is sin u = 5/13 and cos v = -3/5.
Hope this helps.
