Prove that \sin x+\cos x=\sqrt2 \sin (x+\frac{\pi}{4}) \sqrt2(x+\frac{\pi}{4})=\sqrt2(\sin x \cos \frac{\pi}{4}+\cos

agreseza 2021-12-27 Answered
Prove that sinx+cosx=2sin(x+π4)
2(x+π4)=2(sinxcosπ4+cosxsinπ4)=sinx+cosx
Could you solve it from opposite?
sinx+cosx=2sin(x+π4)
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Expert Answer

Paul Mitchell
Answered 2021-12-28 Author has 40 answers
sinx+cosx=212(sinx+cosx)
=2(12sinx+12cosx)
=2(sinxcos(π4)+sin(π4)cosx)
Hint: sinAcosB+sinBcosA=sin(A+B)
=2sin(x+π4)
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yotaniwc
Answered 2021-12-29 Author has 34 answers

From
sin(x+y)=sinxcosy+cosxsiny (1)
and
sin(y)=siny (2)
(both of which identities I assume as known and will not prove here), we obtain
sin(xy)=sinxcosycosxsiny (3)
and by adding (1)+(3),
sin(x+y)+sin(xy)=2sinxcosy
Substitue x+yu and xyv, i.e., xu+v2 and yuv2 to arrive at
sinu+sinv=2sinu+v2cosuv2

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nick1337
Answered 2022-01-08 Author has 510 answers

Using the formula sin(x+y)=sinxcosy+cosxsiny
See for first line we know cos(π4) and sin(π4) is equal to 12
So the expression reduces to
2sin(x+π4)=2(sinx2+cosx2)
Now I hope you can reduce it to second line.
For the second line use the formula I gave in starting of answer.

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