Trigonometry & Inverse Trigonometry
Exact Value of Trig Expression — α+β
nta_pyq_2026_jan
Grade 11
Question:
If $\dfrac{\cos^248°-\sin^212°}{\sin^224°-\sin^26°}=\dfrac{\alpha+\beta\sqrt{5}}{2}$, where $\alpha,\beta\in\mathbb{N}$, then $\alpha+\beta$ is equal to _____.
Step-by-Step Solution
Key Concept: $\cos^248°-\sin^212°=\cos60°\cos36°=\tfrac{1}{2}\cos36°$. $\sin^224°-\sin^26°=\sin30°\sin18°=\tfrac{1}{2}\sin18°$. Ratio $=\dfrac{\cos36°}{\sin18°}$.
$\alpha=3$, $\beta=1$. $\alpha+\beta=4$.
Correct Answer: 4