Trigonometry & Inverse Trigonometry
Trigonometry
star_batch_jee_advanced_2025
Grade 11
Question:
If $\sum_{r=1}^{q}\frac{\tan 2^{r-1}}{\cos 2^r} = \tan p^n - \tan q$, then find the value of $(p + q)$
Step-by-Step Solution
Key Concept: Telescoping sums arise from expressing $\frac{\sin(A-B)}{\cos A \cos B} = \tan A - \tan B$.
The sum $\sum_{r=1}^{n} \frac{\sin(2^r - 2^{r-1})}{\cos 2^r \cos 2^{r-1}}$ telescopes by rewriting each term as $\tan 2^r - \tan 2^{r-1}$. When summed from $r=1$ to $n$, intermediate terms cancel, leaving $\tan 2^n - \tan 1$.
Correct Answer: 3