Trigonometry
Telescoping product using sin difference identity
MMTS_Full_Test_04
Grade 12
Question:
Let $x=\sin1°$. The value of $\dfrac{1}{\cos0°\cos1°}+\dfrac{1}{\cos1°\cos2°}+\cdots+\dfrac{1}{\cos44°\cos45°}$ is
(A) $x$
(B) $1/x$
(C) $\sqrt2/x$
(D) $x/\sqrt2$
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: (B) $1/x$