Answer :

Given,



To prove: sinα + cosα = √2 cos θ




{ tan A = (sinA)/(cosA)}


{ tan π/4 = 1}


We know that: tan(x-y) =



θ = α - π/4


α = θ + π/4 …(1)


As we have to prove - sinα + cosα = √2 cos θ


LHS = sinα + cosα


LHS = sin(θ + π/4) + cos(θ + π/4) {using equation 1}


sin(x + y) = sin x cos y + cos x sin y


And, cos(x + y) = cos x cos y – sin x sin y


LHS = sin θ cos(π/4) + sin(π/4)cos θ + cos θ cos(π/4) - sin(π/4)sin θ


sin(π/4)=cos(π/4) = 1/√2


LHS =


LHS =


LHS = √2 cos θ = RHS


Hence, sinα + cosα = √2 cos θ


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