Question

In an equiliteral spherical triangle ABC show that secA =1+seca

Answer 1

Answer:

So A is the vertex or the angle of the triangle and a is the side of the equilateral spherical triangle.

I started off the proof by using the law of cosine:

cos(a)−cos(a)cos(a)=sin(a)sin(a)cos(A)cos⁡(a)−cos⁡(a)cos⁡(a)=sin⁡(a)sin⁡(a)cos⁡(A)

and after simplifying it a bit, I obtained:

cos(a)−cos2(a)=sin2(a)cos(A)cos⁡(a)−cos2⁡(a)=sin2⁡(a)cos⁡(A)

I replaced sin2(a)sin2⁡(a) with (1−cos2(a))(1−cos2⁡(a)).

Then I obtained:

cos(a)−cos2(a)=(1−cos2(a))cos(A)cos⁡(a)−cos2⁡(a)=(1−cos2⁡(a))cos⁡(A)

and I realized on the left side, I can pull a cos(a)cos⁡(a). So through factoring:

cos(a)(1−cos(a))=(1−cos2(a))cos(A)cos⁡(a)(1−cos⁡(a))=(1−cos2⁡(a))cos⁡(A)

Either I'm not seeing it but I do not how to proceed after this. If anyone can help, I'd like that. Thanks