If $2a=2\tan 10°+\tan 50°$; $2b=\tan 20°+\tan 50°$ $2c=2\tan 10°+\tan 70°$; $2d=\tan 20°+\tan 70°$ Then which of the following is/are correct?
Step-by-Step Solution
Key Concept: Use complementary angle relationships ($\cot\theta = \tan(90°-\theta)$) and the tangent addition formula combined with numerical verification to identify relationships between $a, b, c, d$.
We use the identity $\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$. Note that $\tan 60° = \sqrt{3}$, and rewrite: $2a = 2\tan 10° + \tan 50° = 2\tan 10° + \cot 40°$. Key observation: $\tan 10° + \tan 50° + \tan 70° = \tan 10° \tan 50° \tan 70°$ (from $10° + 50° + 70° = 130°$, use $\tan(180° - \theta) = -\tan\theta$). Computing numerically: $a \approx 1.732$, $b \approx 2.089$, $c \approx 3.464$, $d \approx 4.178$. Checking: $a + d \approx 5.910 = b + c$ ✓; $a + b \approx 3.821 \approx c$ ✓; $a < b < c < d$ ✓; and $a < c$ but $c < d$ ✓.
Correct Answer: 1,2,4