Question

Without using tables, evaluate the following: cosec²45° sec²30° - sin²30° - 4cot²45° + sec²60°.

Answer 1

cosec²45° sec²30° - sin²30° - 4cot²45° + sec²60°.

sin45° = `(1)/sqrt(2)`

cosec45° = `sqrt(2)/(1)`

sin30° = `(1)/(2)` = cos60°

sec60° = 2

cos30° = `sqrt(3)/(2)`

sec30° = `(2)/sqrt(3)`

tan45° = 1
cot45° = 1
cosec²45° sec²30° - sin²30° - 4cot²45° + sec²60°

= `(sqrt(2)/1)^2 (2/sqrt(3))^2 - (1/2)^2 - 4(1)^2 + (2)^2`

= `2 xx (4)/(3) - (1)/(4) - 4 + 4`

= `(8)/(3) - (1)/(4)`

= `(32 - 3)/(12)`

= `(29)/(12)`.