- #1
Albert1
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prove:
$tan 1^o+tan 5^o+tan 9^o +---------+tan 177^o=45$
$tan 1^o+tan 5^o+tan 9^o +---------+tan 177^o=45$
perfect (Yes)Opalg said:Outline solution:
[sp]For $0\leqslant k\leqslant 44$, the angles $\theta = (4k+1)^\circ$ satisfy $\tan(45\theta) = 1.$
The formula for $\tan(n\theta)$ gives $\tan(45\theta) = \dfrac{{45\choose1}t - {45\choose3}t^3 + \ldots -{45\choose43}t^{43} + t^{45}}{1 - {45\choose2}t^2 - \ldots - {45\choose44}t^{44}} = \dfrac{45t -\ldots + t^{45}}{1-\ldots + 45t^{44}},$ where $t = \tan\theta.$ So the equation $\tan(45\theta) = 1$ (for $\theta$) corresponds to the equation $\dfrac{45t -\ldots + t^{45}}{1-\ldots + 45t^{44}} = 1$ (for $t$), or equivalently $t^{45} - 45t^{44} - \ldots -1=0.$ The sum of the roots of that equation is $45.$ Therefore \(\displaystyle \sum_{k=0}^{44}\tan(4k+1)^\circ = 45.\)[/sp]
The purpose of proving trigonometric equality is to verify the relationship between different trigonometric functions and angles, and to demonstrate that they are equivalent under certain conditions.
To prove a trigonometric equality, one must use mathematical techniques such as substitution, simplification, and manipulation of identities to transform the given equation into a form that is easier to work with.
The key identities used in proving trigonometric equality include the Pythagorean identities, sum and difference identities, double angle identities, and half angle identities.
To prove this equality, we can use the sum and difference identities for tangent and the fact that tan 45^o = 1. By substituting the values, we can simplify the equation and show that both sides are equal to 1.
Proving trigonometric equality is important in mathematics and science because it allows us to accurately calculate and analyze geometric and trigonometric problems, and to understand the relationships between different trigonometric functions and angles.