Question

化简sinα+cosα/cos（α-π/4）

Answer 1

simplify:

(sin(a)*sin(a+1/4*pi)+cos(a))/sin(a+1/4*pi)

radsimp:

(sin(a)*sin(a+1/4*pi)+cos(a))/sin(a+1/4*pi)

combine(trig):

1/4*(2^(1/2)-2*cos(2*a+1/4*pi)+4*cos(a))/sin(a+1/4*pi)

factor:

(sin(a)*sin(a+1/4*pi)+cos(a))/sin(a+1/4*pi)

expand:

sin(a)+cos(a)/(1/2*sin(a)*2^(1/2)+1/2*cos(a)*2^(1/2))

combine:

1/4*(2^(1/2)-2*cos(2*a+1/4*pi)+4*cos(a))/sin(a+1/4*pi)

convert(exp):

-1/2*i*(exp(i*a)-1/exp(i*a))+2*i*(1/2*exp(i*a)+1/2/exp(i*a))/(exp(i*(a+1/4*pi))-1/exp(i*(a+1/4*pi)))

convert(sincos):

sin(a)+cos(a)/sin(a+1/4*pi)

convert(tan):

2*tan(1/2*a)/(1+tan(1/2*a)^2)+1/2*(1-tan(1/2*a)^2)/(1+tan(1/2*a)^2)/tan(1/2*a+1/8*pi)*(1+tan(1/2*a+1/8*pi)^2)

collect(pi):

sin(a)+cos(a)/sin(a+1/4*pi)

通过软件分解，这已经是最简形式。

Answer 2

首先，如果不是题主写错，原题为（sinα+cosα）/cos（α-π/4）的话我无法化简
其次，（sinα+cosα）/cos（α-π/4）=（sinα+cosα）/（cosαcosπ/4+sinαsinπ/4）=（sinα+cosα）/√2/2(cosα+sinα)=√2