Trigonometry
Telescoping product using sin difference identity
Grade Class 12
Question:
Let $x=\sin1°$. The value of $\dfrac{1}{\cos0°\cos1°}+\dfrac{1}{\cos1°\cos2°}+\cdots+\dfrac{1}{\cos44°\cos45°}$ is
$x$
$1/x$
$\sqrt2/x$
$x/\sqrt2$
Step-by-Step Solution
Key Concept: $\dfrac{1}{\cos n°\cos(n+1)°}=\dfrac{\tan(n+1)°-\tan n°}{\sin1°}$. Sum telescopes to $\dfrac{\tan45°-\tan0°}{\sin1°}=\dfrac{1}{\sin1°}=1/x$.
$1/x=1/\sin1°$.
Correct Answer: 2