Prove that sin⁡x+cos⁡x=2sin⁡(x+π4) 2(x+π4)=2(sin⁡xcos⁡π4+cos⁡xsin⁡π4)=sin⁡x+cos⁡x Could you solve it from opposite? sin⁡x+cos⁡x=2sin⁡(x+π4)

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Prove that sin⁡x+cos⁡x=2sin⁡(x+π4)  2(x+π4)=2(sin⁡xcos⁡π4+cos⁡xsin⁡π4)=sin⁡x+cos⁡x  Could you solve it from opposite?  sin⁡x+cos⁡x=2sin⁡(x+π4)

Paul Mitchell · Accepted Answer

sin⁡x+cos⁡x=212(sin⁡x+cos⁡x)  =2(12sin⁡x+12cos⁡x)   =2(sin⁡xcos⁡(π4)+sin⁡(π4)cos⁡x)   Hint: sin⁡Acos⁡B+sin⁡Bcos⁡A=sin⁡(A+B)   =2sin⁡(x+π4)

yotaniwc · Accepted Answer

From sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y (1) and sin⁡(−y)=−sin⁡y (2) (both of which identities I assume as known and will not prove here), we obtain sin⁡(x−y)=sin⁡xcos⁡y−cos⁡xsin⁡y (3) and by adding (1)+(3), sin⁡(x+y)+sin⁡(x−y)=2sin⁡xcos⁡y Substitue x+y←u and x−y←v, i.e., x←u+v2 and y←u−v2 to arrive at sin⁡u+sin⁡v=2sin⁡u+v2cos⁡u−v2

nick1337 · Accepted Answer

Using the formula sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y See for first line we know cos⁡(π4) and sin⁡(π4) is equal to 12 So the expression reduces to 2sin⁡(x+π4)=2(sin⁡x2+cos⁡x2) Now I hope you can reduce it to second line. For the second line use the formula I gave in starting of answer.

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

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