Given sinθ + cosθ = cos 2θ, show that tanθ = −1 or cosθ − sinθ = 1

[hollicrombie&stitch]

These are the methods I tried already:
--> sinθ + cosθ = cos²θ - sin²θ then using t-formula to derive an expression for tan(θ/2) (was unable to reach a statement where t=something, also in the textbook this is the exercise before the one on t-formula so it shouldn't be necessary to solve it)
--> converting cos 2θ into 1 - 2sin²θ then trying to divide it all out by cosθ to get an expression in terms of tanθ (I ended up with secθ and a leftover sinθ that didn't do me any wonders)

I am able to successfully solve for θ in the domain [0,2π] by using auxiliary angles to convert sinθ + cosθ into √2 cos(θ-π/4) but cannot show the identities in the title. Don't need a step-by-step walk through of the whole problem, just a push in the right direction (first step or two). Thanks!

 

[Deleted member 4993]

These are the methods I tried already:
--> sinθ + cosθ = cos²θ - sin²θ then using t-formula to derive an expression for tan(θ/2) (was unable to reach a statement where t=something, also in the textbook this is the exercise before the one on t-formula so it shouldn't be necessary to solve it)
--> converting cos 2θ into 1 - 2sin²θ then trying to divide it all out by cosθ to get an expression in terms of tanθ (I ended up with secθ and a leftover sinθ that didn't do me any wonders)

I am able to successfully solve for θ in the domain [0,2π] by using auxiliary angles to convert sinθ + cosθ into √2 cos(θ-π/4) but cannot show the identities in the title. Don't need a step-by-step walk through of the whole problem, just a push in the right direction (first step or two). Thanks!

cos(2t) = [cos(t) + sin(t)] * [cos(t) - sin(t)]

sin(t) + cos(t) = [cos(t) + sin(t)] * [cos(t) - sin(t)]

[sin(t) + cos(t)] * [1 - {cos(t) - sin(t)}] = 0

continue....

 

[hollicrombie&stitch]

Thank-you! I got cosθ - sinθ = 1 but still can't seem to show tanθ = -1

 

[Deleted member 4993]

Thank-you! I got cosθ - sinθ = 1 but still can't seem to show tanθ = -1

That you got from analyzing the "second" factor.

What do you get when you analyze the first factor?

 

[hollicrombie&stitch]

Got it! Apologies; I didn't use your final line of working "[sin(t) + cos(t)] * [1 - {cos(t) - sin(t)}] = 0" but rather went from the second last line and equated the coefficients of [cos(t) + sin(t)] on each side of the equation, giving cosθ - sinθ = 1.
Thanks for all the help!