Let $\triangle ABC$ be a right spherical triangle such that the angle $\sphericalangle C$ is a right angle. :500px Let the remaining parts of $\triangle ABC$ be arranged according to the '''interior''' 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 for angle $A$ }} {{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 {\Box - c} \tan b       | c = Sine of Complement equals Cosine, Tangent of Complement equals Cotangent }} {{end-eqn}} {{qed|lemma}} ==== $\map \sin {\Box - c}$ ==== {{begin-eqn}} {{eqn | l = \cos C       | r = -\cos A \cos B + \sin A \sin B \cos c       | c = Spherical Law of Cosines for angle $C$ }} {{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 {\Box - c}       | r = \map \tan {\Box - A} \, \map \tan {\Box - B}        | c = Sine of Complement equals Cosine, 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 for angle $B$ }} {{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 {\Box - c} \tan a       | c = Sine of Complement equals Cosine, Tangent of Complement equals Cotangent }} {{end-eqn}} {{qed}}