So, the equation is sin(x + y)/sin(x - y) = (1 + cotxstany)/(1 + cotxtany)
The question has to to with trigonometric identities?
What are trigonometric identities?
Trigonometric identities are equations that show the relationship between the trigonometric ratios.
How to solve the equation?
Given the equation sin(x + y)/sin(x - y)
Using the trigonometric identities.
- sin(x + y) = sinxcosy + cosxsiny and
- sin(x - y) = sinxcosy - cosxsiny
So, sin(x + y)/sin(x - y) = (sinxcosy + cosxsiny)/(sinxcosy + cosxsiny)
Dividing the rnumerator and denominator of ight hand side by sinx, we have
sin(x + y)/sin(x - y) = (sinxcosy + cosxsiny)/sinx/(sinxcosy + cosxsiny)/sinx
sin(x + y)/sin(x - y) = (sinxcosy/sinx + cosxsiny/sinx)/(sinxcosy/sinx + cosxsiny/sinx)
= (cosy + cotxsiny)/(cosy + cotxsiny) (since cosx/sinx = cotx)
Dividing the numerator and denominator of the right hand side by cosy, we have
= (cosy + cotxsiny)/cosy/(cosy + cotxsiny)/cosy
= (cosy/cosy + cotxsiny/cosy)/(cosy/cosy + cotxsiny/cosy)
= (1 + cotxstany)/(1 + cotxtany) [since siny/cosy = tany]
So, sin(x + y)/sin(x - y) = (1 + cotxstany)/(1 + cotxtany)
Learn more about trigonometric identities here:
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