If x- A-cos A and y- A-cos A- prove that x-y x-y 2-x-y 22-1

LHS=x yx+y2+x y22=cotA+cosA cotA cosAcotA+cosA+cotA cosA2+cotA+cosA cotA cosA22=cotA+cosA cotA+cosAcotA+cosA+cotA cosA2+cotA+cosA cotA+cosA22=2cosA2cotA2+2cosA2 ...

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Standard X

Mathematics

Complementary Trigonometric Ratios

If x= A+cos A...

Question

If $x = cotA + cosA$ and $y = cotA - cosA$ , prove that ${(\frac{x - y}{x + y})}^{2} + {(\frac{x - y}{2})}^{2} = 1$ .

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Solution

$LHS = {(\frac{x - y}{x + y})}^{2} + {(\frac{x - y}{2})}^{2} = {[\frac{(cotA + cosA) - (cotA - cosA)}{(cotA + cosA) + (cotA - cosA)}]}^{2} + {[\frac{(cotA + cosA) - (cotA - cosA)}{2}]}^{2} = {[\frac{cotA + cosA - cotA + cosA}{cotA + cosA + cotA - cosA}]}^{2} + {[\frac{cotA + cosA - cotA + cosA}{2}]}^{2} = {[\frac{2 cosA}{2 cotA}]}^{2} + {[\frac{2 cosA}{2}]}^{2} = {[\frac{cosA}{(\frac{cosA}{sinA})}]}^{2} + {[cosA]}^{2} = {[\frac{sinA cosA}{cosA}]}^{2} + {[cosA]}^{2} = {[sinA]}^{2} + {[cosA]}^{2} = \sin^{2} A + \cos^{2} A = 1 = RHS$

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If x=cotA+cosA and y=cotA-cosA, prove that x-yx+y2+x-y22=1.

LHS=x-yx+y2+x-y22=cotA+cosA-cotA-cosAcotA+cosA+cotA-cosA2+cotA+cosA-cotA-cosA22=cotA+cosA-cotA+cosAcotA+cosA+cotA-cosA2+cotA+cosA-cotA+cosA22=2cosA2cotA2+2cosA22=cosAcosAsinA2+cosA2=sinA cosAcosA2+cosA2=sinA2+cosA2=sin2A+cos2A=1=RHS

If $x = cotA + cosA$ and $y = cotA - cosA$ , prove that ${(\frac{x - y}{x + y})}^{2} + {(\frac{x - y}{2})}^{2} = 1$ .