What is sec squared 45 degrees plus csc squared 60 degrees?

Answer 1

You use the definitions of secant (sec) and cosecant (csc):

sec x = 1/cos x

csc x = 1/sin x

→ sec² 45° + csc² 60°

= (1/cos 45°)² + 1/(sin 60°)²

= (1/(1/√2))² + (1/(√3/2))²

= (√2)² + (2/√3)²

= 2 + 4/3

= 10/3 = 3⅓

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In a right-triangle with an angle of 45°, the other angle is 45° and the sides are in the ratio of 1 : 1 : √2

(sides opposite angles 45° : 45° : 90°) giving cos 45° = 1/√2

In a right triangle with an angle of 60°, the other angle is 30° and the sides are in the ratio of √3 : 1 : 2 (sides opposite angles 60° : 30° : 90°) giving sin 60° = √3/2

These ratios can be confirmed/calculated by Pythagoras; in the first case it is an isosceles triangle with legs of length 1 unit, and in the second case it is half an equilateral triangle (one angle bisected by a perpendicular from the opposite side) with side length 2 units.