If x-cosec A-cos A and y-cosec A-cos A- then prove that 2 x-y 2-x-y 22-1-0

LHS=2x+y2+x y22 1=2cosecA+cosA+cosecA cosA2+cosecA+cosA cosecA cosA22 1=2cosecA+cosA+cosecA cosA2+cosecA+cosA cosecA+cosA22 1=22cosecA2+2cosA22 1 =1cosecA2+cosA ...

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Byju's Answer

Standard XII

Mathematics

Implicit Differentiation

If x=cosec A+...

Question

If $x = cosecA + cosA$ and $y = cosecA - cosA$ , then prove that ${(\frac{2}{x + y})}^{2} + {(\frac{x - y}{2})}^{2} - 1 = 0$ .

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Solution

$LHS = {(\frac{2}{x + y})}^{2} + {(\frac{x - y}{2})}^{2} - 1 = {[\frac{2}{(cosecA + cosA) + (cosecA - cosA)}]}^{2} + {[\frac{(cosecA + cosA) - (cosecA - cosA)}{2}]}^{2} - 1 = {[\frac{2}{cosecA + cosA + cosecA - cosA}]}^{2} + {[\frac{cosecA + cosA - cosecA + cosA}{2}]}^{2} - 1 = {[\frac{2}{2 cosecA}]}^{2} + {[\frac{2 cosA}{2}]}^{2} - 1$
$= {[\frac{1}{cosecA}]}^{2} + {[cosA]}^{2} - 1 = {[sinA]}^{2} + {[cosA]}^{2} - 1 = \sin^{2} A + \cos^{2} A - 1 = 1 - 1 = 0 = RHS$

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If x=cosecA+cosA and y=cosecA-cosA, then prove that 2x+y2+x-y22-1=0.

LHS=2x+y2+x-y22-1=2cosecA+cosA+cosecA-cosA2+cosecA+cosA-cosecA-cosA22-1=2cosecA+cosA+cosecA-cosA2+cosecA+cosA-cosecA+cosA22-1=22cosecA2+2cosA22-1 =1cosecA2+cosA2-1=sinA2+cosA2-1=sin2A+cos2A-1=1-1=0=RHS

If $x = cosecA + cosA$ and $y = cosecA - cosA$ , then prove that ${(\frac{2}{x + y})}^{2} + {(\frac{x - y}{2})}^{2} - 1 = 0$ .