Question

If sinθ + cosθ = √2cosθ, (θ ≠ 90o) then the value of tanθ is

Answer 1

If sinθ + cosθ = √2cosθ  then the value of tanθ is √2 − 1

Step-by-step explanation:

sinθ + cosθ = √2cosθ

Squaring both sides

sin²θ + cos²θ + 2sinθcosθ= 2cos²θ

using sin²θ + cos²θ = 1  & Sin2θ = 2sinθcosθ

=> 1 + Sin2θ = 2cos²θ

=> Sin2θ = 2cos²θ -1

using Cos2θ = 2cos²θ -1

=>  Sin2θ = Cos2θ

=>  Sin2θ/Cos2θ = 1

=> Tan2θ = 1

now using

Tan2θ = 2Tanθ/(1  - Tan²θ)

=> 1 = 2Tanθ/(1  - Tan²θ)

=> 1  - Tan²θ = 2Tanθ

=> Tan²θ + 2Tanθ - 1 = 0

=> Tanθ  =  (-2 ± √(4 + 4) )/2

=> Tanθ  = -1 ± √2

hence Tanθ  = √2 − 1   ( from the given option)

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