Trigonometry & Inverse Trigonometry
Sum of trigonometric ratios; value of an expression
nta_pyq_2023_jan
Grade 11
Question:
If $\tan 15° + \frac{1}{\tan 75°} + \frac{1}{\tan 105°} + \tan 195° = 2a$, then the value of $\left(a + \frac{1}{a}\right)$ is:
4
$4 - 2\sqrt{3}$
2
$5 - \frac{3}{2}\sqrt{3}$
Step-by-Step Solution
Key Concept: Use exact values: $\tan 15° = 2-\sqrt{3}$, $\cot 75° = 2-\sqrt{3}$, $\cot 105° = -(\sqrt{3}-2) = 2-\sqrt{3}$ ... wait $\cot 105° = \sqrt{3}-2$. Sum and find $a$.
$a = 2-\sqrt{3}$. $a + 1/a = 2-\sqrt{3} + \frac{1}{2-\sqrt{3}} = 2-\sqrt{3} + (2+\sqrt{3}) = 4$. Answer: (1)
Correct Answer: 4