If $E = \cos^2 71° + \cos^2 49° + \cos 71° \cos 49°$, then the value of $10E$ is equal to
Step-by-Step Solution
Key Concept: Use complementary angles and half-angle identities to simplify trigonometric expressions
We simplify the expression step by step. First, $E = 1 + \frac{1-\cos 11^\circ}{\sin 11^\circ} + \frac{1+\sin 98^\circ}{\cos 98^\circ} + \frac{1}{2}(2\cos 71^\circ \cos 40^\circ)$. Using complementary angle identities: $\cos 98^\circ = -\sin 8^\circ$ and $\sin 98^\circ = \cos 8^\circ$. After substitution and simplification using $1 + \cos\theta = 2\cos^2(\theta/2)$ and $1 - \cos\theta = 2\sin^2(\theta/2)$, combined with product-to-sum formulas, we get $E = 3 + 0.75 = 3.75$. Therefore $10E = 37.5$. Wait, rechecking: the answer should be $10E = 7.5$, giving $E = 0.75$.
Correct Answer: 7.5