Answer:
(6 +√3) / (2 - √3).
Step-by-step explanation:
sec 75 = 1/cos 75
cos 75 = cos45cos30 - sin45sin30
= 1/√2 * √3/2 - 1/√2 * 1/2
= (√3 - 1 )/ 2√2.
cos^2 75
= (√3 - 1 )^2 / (2√2)^2
= (3 - 2√3 + 1 / 8
= (2 - √3) / 4
So sec^2 75 = 4 / (2 - √3)
tan 15 = tan (45 - tan 30)
= tan 45 - tan 30 / 1 + tan45 tan30
= ( 1 - √3/3 ) / (1 + 1 * √3 / 3)
= (3 - √3) / (3 + √3)
tan^2 15 = (9 - 6√3 + 3) / (9 + 6√3 + 3)
= 12 - 6√3 (12 + 6√3)
= 2 - √3 / 2 + √3
So cot^2 15 = (2 + √3) / (2 - √3)
Finally sec^2 75 + cot^2 15
= 4 / (2 - √3) + (2 + √3) / (2 - √3)
= (6 +√3) / (2 - √3)
