plzzzzz solve this fast
Answers
Answer:
(2) √3
Step-by-step explanation:
Looking at the options, it looks worthwhile to square the expression first and see if we get 1/3, 3, 12 or 1/4.
To keep things tidy, let's write c = cos 70° and s = sin 70°.
We need:
- cot 70° = cos 70° / sin 70° = c/s
- cos²70° + sin²70° = 1 => c² = 1 - s²
- sin 3A = 3 sin A - 4 sin³ A
This last identity is useful because we need a way to relate 70° to a familiar angle like 30°, 45° or 60°. We can do this by noticing that
3 × 70° = 210° = 180° + 30°
So
3s - 4s³ = sin(3 × 70°) = sin(180° + 30°) = - sin 30° = -1/2
=> 8s³ = 1 + 6s
Now to work...
The square of the given expression is:
( cot 70° + 4 cos 70° )²
= ( c/s + 4c )²
= c² ( 1 + 4s )² / s²
= ( 1 - s² ) ( 1 + 8s + 16s² ) / s²
= ( 1 + 8s + 15s² - 8s³ - 16s⁴ ) / s²
= ( 1 + 8s + 15s² - ( 1 + 6s ) - 2s ( 1 + 6s ) ) / s²
= ( 1 + 8s + 15s² - 1 - 6s - 2s + 12s² ) / s²
= 3s² / s²
= 3
Since cot 70° and cos 70° are both positive, it follows that
( cot 70° + 4 cos 70° ) = √3.