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sina +2cosa equal to 1 then prove 2sina -cosa equal to 2
Asked by hakushwah | 13 Jan, 2019, 11:23: PM Expert Answer
Given :  sinA+2cosA = 1   ....................................(1)

Required to prove : 2sinA - cosA = 2  ...........................(2)

let us use the notation, s = sinA and c = cosA

Then given trignometric equation is written as,  s+2c = 1  .................(3)

let us write eqn.(3) as :    s = 1 - 2c
squaring both sides of the above eqn.,  we have   s²= (1-2c)² = 1 + 4c² -4c

4c²-4c+(1-s²) = 0    or  5c² - 4c = 0   ( because 1-sin2θ = cos²θ )

hence we have,   c(5c-4) = 0   ; hence c = 0 or c=4/5 .............................(4)

if we consider c = cosA = 4/5 , then sinA = hence required eqn. :-   if sinA = 3/5 and cosA=4/5,  then 2sinA - cosA = (6/5)-(4/5) = 2/5  ≠ 2

if sinA = -3/5 and cosA=4/5, then 2sinA - cosA = (-6/5)-(4/5)  = -2  ≠ 2

in eqn.(4) if we consider c = cosA = 0, then sinA = 1

hence required eqn. :-   2sinA - cosA =  2(1) - 0 = 2

Conclusion : if cosA=0 as we get in eqn.(4), then 2sinA - cosA = 2
Answered by Thiyagarajan K | 14 Jan, 2019, 08:21: AM

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