<prompt> Kokotakushoto 1 - frac1cos 23circ kulia kushoto 1 frac1sin 67circ kulia kushoto 1 - frac1sin 23circ kulia kushoto 1 frac1cos 67circ kulia.  <response>: Ninatambua kuwa digrii 67 ni kikamilisho cha digrii 23, kwa hivyo ninaweza kutumia kitambulisho sin90circ - theta cos theta na kinyume chake. Hii inamaanisha kuwa naweza kuandika upya usemi kama left 1 - frac1cos 23circ kulia kushoto 1 frac1cos 23circ kulia kushoto 1 - frac1sin 23circ kulia kushoto 1 frac1sin 23circ kulia. Sasa naona kwamba kila jozi ya mambo katika mabano ni ya umbo 1 - frac1x na 1 frac1x, ambayo huzidisha hadi 1 - frac1x2 . Kwa hivyo usemi unarahisisha kuwa left 1 - frac1cos2 23circ kulia left 1 - frac1sin2 23circ kulia. Sasa nakumbuka kitambulisho cha Pythagorean sin2 theta cos2 theta 1, ambayo inamaanisha kuwa frac1cos2 theta 1 tan2 theta na frac1sin2 theta 1 cot2 theta. Nikibadilisha hizi kwenye usemi, napata left 1 - frac1cos2 23circ kulia kushoto 1 - frac1sin2 23 circ kulia kushoto 1 - 1 tan2 23circ kulia kushoto 1 - 1 cot2 23circ kulia - tan2 23circ -cot2 23circ. Hatimaye, ninatumia kitambulisho tan theta frac1cot theta kurahisisha kujieleza zaidi: -tan2 23circ -cot2 23circ -tan2 23circ left -frac1 tan2 23circ kulia 1. 