Verify the identity 5csc2 alpha 8cot alpha 263 csc2 alpha

Verify the identity 5csc^2 alpha - 8cot alpha - 26/3 csc^2 alpha - 13 cot alpha + 9 = 5 cot alpha+7/3 cot alpha - 4

Solution

[5csc²a -8cota -26]/[3csc²a -13cota +9]

csc²a -cot²a=1 is identity =>csc²a=1+cot²a

=>[5(1+cot²a) -8cota -26]/[3(1+cot²a) -13cota +9]

=>[5+5cot²a -8cota -26]/[3+3cot²a -13cota +9]

=>[5cot²a -8cota -21]/[3cot²a -13cota +12]

=>[5cot²a -15cota+7cota -21]/[3cot²a -9cota-4cota +12]

=>[5cota(cota -3)+7(cota -3)]/[3cota(cota -3)-4(cota-3)]

=>[(cota -3)(5cota+7)]/[(cota -3)(3cota-4)]

=>(5cota+7)/(3cota-4)