Let $\triangle ABC$ be a quadrantal triangle on the surface of a sphere whose center is $O$ such that side $c$ is a right angle. :500px Let the remaining parts of $\triangle ABC$ be arranged according to the '''exterior''' of the circle above, where the symbol $\Box$ denotes a right angle. ==== $\sin A$ ==== {{begin-eqn}} {{eqn | l = \cos A \cos c       | r = \sin A \cot B - \sin c \cot b       | c = Four-Parts Formula on $b, A, c, B$ }} {{eqn | ll= \leadsto       | l = \cos A \times 0       | r = \sin A \cot B - 1 \times \cot b       | c = Cosine of Right Angle, Sine of Right Angle as $c = \Box$ }} {{eqn | ll= \leadsto       | l = \sin A \cot B       | r = \cot b       | c =  }} {{eqn | ll= \leadsto       | l = \sin A       | r = \tan B \cot b       | c = multiplying both sides by $\tan B = \dfrac 1 {\cot B}$ }} {{eqn | ll= \leadsto       | l = \sin A       | r = \tan B \, \map \tan {\Box - b}       | c = Tangent of Complement equals Cotangent }} {{end-eqn}} {{qed|lemma}} ==== $\sin B$ ==== {{begin-eqn}} {{eqn | l = \cos B \cos c       | r = \sin B \cot A - \sin c \cot a       | c = Four-Parts Formula on $A, c, B, a$ }} {{eqn | ll= \leadsto       | l = \cos B \times 0       | r = \sin B \cot A - 1 \times \cot a       | c = Cosine of Right Angle, Sine of Right Angle as $c = \Box$ }} {{eqn | ll= \leadsto       | l = \sin B \cot A       | r = \cot a       | c =  }} {{eqn | ll= \leadsto       | l = \sin B       | r = \tan A \cot a       | c = multiplying both sides by $\tan A = \dfrac 1 {\cot A}$ }} {{eqn | ll= \leadsto       | l = \sin B       | r = \tan A \, \map \tan {\Box - a}       | c = Tangent of Complement equals Cotangent }} {{end-eqn}} {{qed|lemma}} ==== $\map \sin {\Box - a}$ ==== {{begin-eqn}} {{eqn | l = \sin A \cos c       | r = \cos C \sin B + \sin C \cos B \cos a       | c = Analogue Formula for Spherical Law of Cosines:Corollary for side $a$ }} {{eqn | ll= \leadsto       | l = \sin A \times 0       | r = \cos C \sin B + \sin C \cos B \cos a       | c = Cosine of Right Angle as $c = \Box$ }} {{eqn | ll= \leadsto       | l = \sin C \cos B \cos a       | r = -\cos C \sin B       | c =  }} {{eqn | ll= \leadsto       | l = \cos a       | r = -\cot C \tan B       | c = dividing both sides by $\sin C \cos B$ }} {{eqn | ll= \leadsto       | l = \map \sin {\Box - a}       | r = \map \tan {C - \Box} \tan b       | c = Sine of Complement equals Cosine, Sine Function is Odd, Tangent of Complement equals Cotangent }} {{end-eqn}} {{qed|lemma}} ==== $\map \sin {C - \Box}$ ==== {{begin-eqn}} {{eqn | l = \cos c       | r = \cos a \cos b + \sin a \sin b \cos C       | c = Spherical Law of Cosines for side $c$ }} {{eqn | ll= \leadsto       | l = 0       | r = \cos a \cos b + \sin a \sin b \cos C       | c = Cosine of Right Angle as $c = \Box$ }} {{eqn | ll= \leadsto       | l = \sin a \sin b \cos C       | r = -\cos a \cos b       | c =  }} {{eqn | ll= \leadsto       | l = -\cos C       | r = \cot a \cot b       | c = dividing both sides by $-\sin a \sin b$ }} {{eqn | ll= \leadsto       | l = \map \sin {C - \Box}       | r = \map \tan {\Box - a} \, \map \tan {\Box - b}        | c = Sine of Complement equals Cosine, Sine Function is Odd, Tangent of Complement equals Cotangent }} {{end-eqn}} {{qed|lemma}} ==== $\map \sin {\Box - b}$ ==== {{begin-eqn}} {{eqn | l = \sin B \cos c       | r = \cos C \sin A + \sin C \cos A \cos b       | c = Analogue Formula for Spherical Law of Cosines:Corollary for side $b$ }} {{eqn | ll= \leadsto       | l = \sin B \times 0       | r = \cos C \sin A + \sin C \cos A \cos b       | c = Cosine of Right Angle as $c = \Box$ }} {{eqn | ll= \leadsto       | l = \sin C \cos A \cos b       | r = -\cos C \sin A       | c =  }} {{eqn | ll= \leadsto       | l = \cos b       | r = -\cot C \tan A       | c = dividing both sides by $\sin C \cos A$ }} {{eqn | ll= \leadsto       | l = \map \sin {\Box - b}       | r = \map \tan {C - \Box} \tan a       | c = Sine of Complement equals Cosine, Sine Function is Odd, Tangent of Complement equals Cotangent }} {{end-eqn}} {{qed}}