A right-angled triangle has three angles, #alpha#, #beta# and #theta#, with one of those being #90^"o"#. Let's say #theta = 90#.

Now, we're trying to prove that #sinalpha=cosbeta#. Since all the angles in a triangle add up to #180#, and one of these angles is already #90#, then we can say that #alpha=90-beta# and #beta = 90-alpha.#

Thus #sinalpha=cos(90-alpha)#. Using the compound angle formula, #cos(a-b)=coscosb+sinasinb#, we can say that #cos(90-alpha)=cosalphacos90+sinalphasin90#.

#cos90=0, sin90=1therefore cosalphacos90-sinalphasin90=0cosalpha+1sinalpha=sinalpha#

#cos(90-alpha)=sinalpha#

#cos(90-alpha)=cosbeta#

#thereforecosbeta=sinalpha# ΟΕΔ