First you want to let alpha=arcsin(-5/13)α=arcsin(−513) and beta=arccos(12/13)β=arccos(1213)
So now we are looking for color(red)cos(alpha+beta)!cos(α+β)!
=>sin(alpha)=-5/13" "⇒sin(α)=−513 and " "cos(beta)=12/13 cos(β)=1213
Recall : cos^2(alpha)=1-sin^2(alpha)=>cos(alpha)=sqrt(1-sin^2(alpha))cos2(α)=1−sin2(α)⇒cos(α)=√1−sin2(α)
=>cos(alpha)=sqrt(1-(-5/13)^2)=sqrt((169-25)/169)=sqrt(144/169)=12/13⇒cos(α)=√1−(−513)2=√169−25169=√144169=1213
Similarly, cos(beta)=12/13cos(β)=1213
=>sin(beta)=sqrt(1-cos^2(beta))=sqrt(1-(12/13)^2)=sqrt((169-144)/169)=sqrt(25/169)=5/13⇒sin(β)=√1−cos2(β)=√1−(1213)2=√169−144169=√25169=513
=>cos(alpha+beta)=cos(alpha)cos(beta)-sin(alpha)sin(beta)⇒cos(α+β)=cos(α)cos(β)−sin(α)sin(β)
Then substitue all the values obtained ealier.
=>cos(alpha+beta)=12/13*12/13-(-5/13)*5/13=144/169+25/169=169/169=color(blue)1⇒cos(α+β)=1213⋅1213−(−513)⋅513=144169+25169=169169=1
