Simplify the following :

Question

sin(45°+α)-cos(45°+α)/sin(45°+α)+cos(45°+α)

Answer 1

[sin(45+a)-cos(45+a)]/[sin(45+a)+cos(45+a)]

[sin45cosa+cos45sina-(cos45cosa-sin45sina)]/[sin45cosa+cos45sina+cos45cosa-sin45sina]

sin45º=√2/2 and cos45º=√2/2

[√2/2sina+√2/2sina]/[√2/2cosa+√2/2cosa]

√2sina/√2cosa

sina/cosa

tan(a)

 

 

Answer 2

sin(45°+α)-cos(45°+α)/sin(45°+α)+cos(45°+α)=

=|multiply both numerator and denominator by sin(45°+α)-cos(45°+α)|

=[sin(45°+α)-cos(45°+α)]²/[sin²(45°+α)-cos²(45°+α)]

=(1-sin(90°+2α))/[-cos(90°+2α)]

=|use sin(x+y)=sin(x)cos(y)+sin(y)cos(x) and

   cos(x+y)=cos(x)cos(y)-sin(x)sin(y)|=

=(1-cos(2α))/(sin(2α))