my solution:lfdahl said:Show, that $$\tan^2 18^{\circ} \cdot \tan^254^{\circ} \in \Bbb{Q}.$$
Albert said:my solution:
let $y=\tan\, 18^{\circ} \cdot \cot\,36^{\circ}=\dfrac {2cos^218^o-1}{2cos^218^o}$
$=1-\dfrac{1}{2cos^218^o}$=$\dfrac{\sqrt 5}{5}$
so $y^2=\dfrac {1}{5}\in \Bbb{Q}$
(using $sin\,18^o=cos\,72^o=\dfrac {\sqrt 5-1}{4}---(1)$
(1) can be proved easily using geometry ,which I posted long time ago
A rational trigonometric expression is an expression that involves trigonometric functions (such as sine, cosine, tangent, etc.) and rational numbers (numbers that can be expressed as a ratio of two integers). An example of a rational trigonometric expression is tan(2x)/cos(x).
To determine if a rational trigonometric expression is rational, we need to check if all the trigonometric functions involved have angles that can be expressed as rational numbers (such as π/2, π/4, etc.). If all the angles are rational, then the expression is considered rational.
tan^218°⋅tan^254°∈Q to be in Q?When an expression is in Q, it means that the expression is a rational number. In this case, tan^218°⋅tan^254° is a rational number, which means that it can be expressed as a ratio of two integers.
tan^218°⋅tan^254°∈Q?To show that tan^218°⋅tan^254°∈Q, we can use the trigonometric identity tan(2x) = 2tan(x)/(1-tan^2(x)). By substituting 18° and 54° for x and simplifying, we can show that the expression is indeed a rational number.
Knowing if an expression is a rational trigonometric expression can help us in solving trigonometric equations and simplifying trigonometric expressions. It also allows us to better understand the behavior of trigonometric functions and their relationships with rational numbers.