Mahesh raj B.

asked • 07/15/15

if x+y=45degree prove that cot(x+1) cot(y+1) /( cot(x) cot(y) ) =2

want to ask the important question

Stephanie M.

tutor
Are you sure you've written everything correctly? As stated, the problem doesn't seem to be true:
 
 
If we're working in degrees, let x = 30 and y = 15. Then (cot(31)cot(16)) / (cot(30)cot(15)) = 0.898 ≠ 2
 
If we're working in radians, let x = π/6 and y = π/12. Then (cot(1+π/6)cot(1+π/12)) / (cot(π/6)cot(π/12)) = 0.00233 ≠ 2
 
 
Perhaps you meant to write cot(x)+1 and cot(y)+1 instead of cot(x+1) and cot(y+1)?
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07/15/15

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Michael J. answered • 07/15/15

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We substitute  x + y = 45   into the denominator of the equation.  This way, we can have a equation in which all trig functions have an addition/subtraction identity.
 
[cot(x + 1)cot(y + 1)] / [cot(45 - y)cot(45 - x)] = 2
 
 
Since all the trig functions in this equation is cot, we apply the fundamental tangent identity.
 
tan(x - y) = [(tan(x) - tan(y)] / [1 + tan(x)tan(y)]
tan(x + y) = [(tan(x) + tan(y)] / [1 - tan(x)tan(y)]
 
 
If we reciprocate both sides of the identity, we get
 
cot(x - y) = [1 + tan(x)tan(y)] / [tan(x) - tan(y)]   ---->  Use these identities and apply to prove.
cot(x + y) = [1 - tan(x)tan(y)] / [tan(x) + tan(y)] 
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