<prompt> Jumlasum_x244 2dhambixdhambi11 sec x-1 sek x1inaweza kuandikwa kwa fomu sum_n14 -1n fracPhitheta_nPsitheta_n, ambapo Phi,, Psi ni chaguo za kukokotoa za trigonometric na theta_1,, theta_2, , theta_3, , theta_4 ni digrii in 0,45. Pata theta_1 theta_2 theta_3 theta_4.  <response>: Kwa vitambulisho vya jumla vya bidhaa, tunajua kuwa 2sin a sin b cosa-b - cosab, kwa hivyo 2sinxsin1 cos x-1-cosx1: sum_x244 cosx-1 - cosx11 sek x-1 sek x1 sum_x244 cosx-1 - cosx1 frac1cos x1 - frac1cosx-1 sum_x244 fraccos2x-1-1 cosx-1 - fraccos2x1-1cosx1 sum_x244 kushoto fracsin2x1cosx1kulia - kushotofracsin2x-1cosx-1 kuliaJumla ya darubini hizi kwa maneno mengine, tunapopanua jumla, masharti yote ya kati yataghairi hadi -fracsin21cos1 -fracsin 22cos2 fracsin244cos44 fracsin245cos45 . Sasa tunayo masharti manne tunayotaka. Kuna njia kadhaa za kueleza Phi,,Psi kama vitendaji primitive trigonometric; kwa mfano, tukihamisha sin kwa dhehebu, tunaweza kuieleza kama Phix sinx,, Psix cotx. Kwa vyovyote vile, tuna theta_1,theta_2,theta_3,theta_4 1circ,2circ,44circ,45circ , na jibu ni 124445 92. 