Without using tables, evaluate:

(4 tan 60° sec 30° )
+
[(sin 31 sec 59° + cot 59° cot 31°)/
(8 sin² 30°-tan² 45°)]

The "co" in trig functions means "of the complement".

4 tan 60° sec 30°
= 4 sin60°/cos60° * csc60° = 4sec60°
= 8

sin 31° sec 59° + cot 59° cot 31°
= cos59° sec59° + tan31° cot31°
= 1+1

see what you can do with the rest

To evaluate the given expression:

1. Recall the trigonometric value of each angle:
- tan 60° = √3
- sec 30° = 2
- sin 31°, sec 59°, cot 59°, cot 31°, and sin 30° can be approximated using a calculator.

2. Simplify the expression using the values above:
(4 tan 60° sec 30°)
+ [(sin 31° sec 59° + cot 59° cot 31°)/(8 sin² 30° - tan² 45°)]

= (4 * √3 * 2)
+ [(approximated value for sin 31° * approximated value for sec 59° + approximated value for cot 59° * approximated value for cot 31°) /
(8 * approximated value for sin² 30° - tan² 45°)]

3. Continue evaluating the expression using the calculated values.

Note: If you need more precise values for the trigonometric functions, you can use a scientific calculator or an online calculator that supports trigonometric calculations.

tan 60 = sqrt3 / 2

sec 30 = 1/cos 30 = 2/sqrt3
so
(4 tan 60° sec 30° ) = 4 *1 = 4

sin(30+1) / cos (60-1) = [.5 cos 1 + (sqrt 3)/2 sin 1]/[.5cos1+(sqrt 3)/2sin1]
LOL = 1 !!!
ok, your turn :)