Answer:
The expression equals 1 (one)
Step-by-step explanation:
Recall that all these trigonometric values have exact expression in forms of square roots.
There is also a very important rule for the tangent of an addition of angles, which states:
[tex]tan(\alpha +\beta)=\frac{tan(\alpha)+tan(\beta)}{1-tan(\alpha)\,tan(\beta)}[/tex]
We can use this with [tex]\alpha=15^o\,\,\,\,and\,\,\,\beta=30^o[/tex]
[tex]tan(15^o +30^o)=\frac{tan(15^o)+tan(30^o)}{1-tan(15^o)\,tan(30^o)}[/tex]
since tan of 15 degrees plus 30 degrees is tangent of 45 degrees, which we know is exactly 1 (one), we can multiply both sides of the equal sign by the denominator on the right, and then solve for exactly the expression we want to find:
[tex]tan(45^o)=\frac{tan(15^o)+tan(30^o)}{1-tan(15^o)\,tan(30^o)}\\1=\frac{tan(15^o)+tan(30^o)}{1-tan(15^o)\,tan(30^o)}\\1-tan(15^o)\,tan(30^o)=tan(15^o)+tan(30^o)\\1=tan(15^o)\,tan(30^o)+tan(15^o)+tan(30^o)\\tan(15^o)+tan(30^o)+tan(15^o)\,tan(30^o)=1[/tex]
So the answer we are looking for is 1 (one)
