
Diamond Slopes and Speedy Whales
via… One Identity
by
Gregory V. Akulov, teacher,
Luther College High School, Regina, Canada
and
Oleksii V. Akulov, teacher,
High School #328,Kyiv, Ukraine

Diamond Slopes and Speedy Whales via… One Identity
One Identity.$^1$ Show that $\tan^{1} x + \tan^{1} y = 2 \tan^{1} \Large{\frac{s}{p + \sqrt{s^2 + p^2}}},$
where $s = x + y, p = 1  xy.$

Diamond Slopes
If $m$ is the slope of rhombus diagonal drawn in the Cartesian plane (see Figure 1), and $l, n$ are the slopes of its sides, then
\[m = \frac{a}{b \pm \sqrt{a^2 + b^2}},\]
where $a = l + n, b = 1  nl.$ Prove it.
Hint. You may use One Identity. 

A ThreeSpeedyWhales Problem.$^2$
auf Deutsch
en français
Once upon a time three whales were swimming along the northsouth shoreline in the ocean near the island. Children at the island lighthouse observed whales’ motion and drew their positiontime graphs in Cartesian coordinates. They noticed that intersections of three graphs form an isosceles triangle (see Figure 2). If the first whale swam at 7 m/s [N] and the second one at 1 m/s [N], what is the velocity of the third whale?
Hint. You may use Diamond Slopes Rule. 
Solve Equation: $2  2x  2x^2 = x \sqrt{5 + 5 x^2}$.
Hint. Use One Identity to get $3 \tan^{1} x = \tan^{1} 2.$
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$^1$ first appeared in 1999 volume 2 of Matematyka v Shkoli, the journal of the Ministry of Education and Science of Ukraine
$^2$ first appeared in 2010 volume 48:1 of deltak, the journal of the Mathematic Council of the Alberta Teachers' Association
Copyright © March, 2010 by Gregory V. Akulov, Oleksii V. Akulov
