maths trig help please

hi, please could i have some help on this? I'm not sure how they got 1-2sin theta from 4 sin theta cos theta?
question: https://ibb.co/QjfkP4sp
thanks!

they've squared 2sinT + cosT = 1 (T = "theta") then in the term 4sinTcosT they've just replaced cosT = 1 - 2sinT from the original equation.

Just as an aside, you could also go with half angles, especially with the 1 on the right as that cancels with part of the half angle for cos(T)
2sin(T) + cos(T) = 1
4sin(T/2)cos(T/2) - 2sin^2(T/2) = 0
sin(T/2)(2cos(T/2) - sin(T/2)) = 0
so
sin(T/2) = 0
or
tan(T/2) = 2
But if youd squared stuff up and rejected a couple of the answers, that should be ok, though squaring equations isnt usually the way to go because of the extraneous solns. The usual way to do cos(x)+sin(x) is the harmonic identity

I tried the double angle and got 0 but im not sure how they got 2.21 using it? 0+pi is just pi but that wouldn't work? Thanks!

What did you try? Youd only use double angles if you had a product or square of "normal" trig terms.

sorry not double angles, the simple harmonic formula. I got root 5 sin(theta+0.4636)=1 and solved for theta=0 but not sure where 2.21 comes from?
Also, another question I had was how did you recognise that it could be written as in simple harmonic form, is there a trick?
thanks!

When you solve
sin(theta+0.46) = 1/sqrt(5)
you should get two solutions on 0 to 2pi?
Asin(x) + Bcos(x)
can be written in harmonic form using the angle addition identity (you could work back from the ans?). So the trig args are the same "x" and you have the addition of sin() and cos() (scaled by A and B). It should be in your textbook?

The first solution I got when I solved it was 0 then adding pi gives pi but that is invalid because it doesn't work when I sub it back in

sin(theta+0.46) = 1/sqrt(5)
so cast, trig curves, .... give
theta + 0.46 = arcsin(1/sqrt(5)) and pi-arcsin(1/sqrt(5))
....

Yes, solving what you have written gives 0 and pi but not 2.21, as indicated on the ms

they've squared 2sinT + cosT = 1 (T = "theta") then in the term 4sinTcosT they've just replaced cosT = 1 - 2sinT from the original equation.

Yes, solving what you have written gives 0 and pi but not 2.21, as indicated on the ms