Find $S$ if $S=\tan^21^\circ+\tan^23^\circ+\tan^25^\circ+\dots+\tan^289^\circ$.   I tried converting all tangents above $45^\circ$ to cotangents and added them with tangents with the same angle, but that does not give any result nor does it telescope. Tried writing $\tan^2\theta = 1-\frac{\tan\theta}{\tan2\theta}$, but again, no headway. I am trying to relate this problem with complex numbers, and saying roots of a polynomial are of the form $\tan[(2n+1)\frac{\pi}{180}]$, but haven't been able to do so.
123248
Problem:   Find the value of $$\sum_{r=0}^{44} \tan^2(2r+1)$$  Note: The angles here are in degrees.    I don't know how to solve this question because trigonometric simplifications didn't get me anywhere. I think there was a method to solve this question using complex numbers which I no longer remember. Any hint/help will be appreciated.
54168
I tried in this way.$\tan(a+b)=\frac{(\tan a + \tan b)}{1 - \tan a \tan b }$value of $\tan \frac{\pi}{7}$ is coming in decimal.what to do
116568
Determine the smallest positive value of x(in degrees) for which: $\tan(x+100^{\circ}) = \tan(x+50^{\circ})\tan (x)\tan(x-50^{\circ})$  I tried to apply the formula of $\tan(A+B) = \frac{\tan A + \tan B}{1-\tan A \tan B}$ but that led me nowhere resulting in a huge equation.  Please help.
3557
Let $\Gamma$ be a circle that passes through the origin. Show that we can find real numbers $s$ and $t$ such that $\Gamma$ is the graph of $r = 2s \cos (\theta + t).$  I know this has to be converted to a cartesian equation, but how do I do this, and what do I do after?  Thanks
44532
I have to simplify and evaluate this :   $$\cot 70^\circ+4\cos 70^\circ$$  On evaluating it, the answer comes out to be $1.732$, or $\sqrt 3$ .  I tried to get everything in $\sin$ and $\cos$, but it doesn't go any further. Any hints?
111419
I need to find the equation for a circle (mainly its radius) which is tangent to the following three lines:  $y = 0$  $y = \tan(70)x$  $y = -1.428148x + 0.790201$  For the last tangent line equation, I know that it is tangent at the point $(0.371272, 0.259968)$ However, for the other two I do not know the exact point of tangency, only that the circle is tangent to them.
18581
I am a little weak in trigonometry. I have two questions:   Find the value of $\cos 52^{\circ} + \cos 68^{\circ} + \cos 172^{\circ} $ Find the value of $\sin 28^{\circ}+ \cos 17^{\circ} + \cos 28^{\circ} + \sin 17^{\circ} $   I am asking these questions because: 1. I am weak and unable to solve these. 2. I want to know the difference in these questions.
111290
This question was initially posted on SO ().  I'm not sure the answer given there was correct.  I cannot get the results from those expressions to match my CAD model.  The title pretty much sums it up. How do I solve for theta given the following equation.  $$a = b\tan\theta - \frac{c}{\cos\theta}$$  I am not a student and this is not homework. It's been quite a while since I've done any significant trig and I'm out of time to figure this out.
46357
I gave an exercise to a student the other day, thinking I had a simple solution for it, but it seems that my solution was just bullshit (at least, not conclusive). Well, at least, I warned her I wasn't sure of my solution :-)  Here it is : $a\in[-1,1]$, show that all roots of polynomial $P=X^{n+1}-aX^n+aX-1$ have modulus $1$ (I try different ways of expressing the problem, tell me which one is better english...).  It is easy to see that    roots of $P$ are $1$, $-1$ for some special cases, and complex not real roots $z_i$ such that $\overline{z_i}$ is also a root, as $P$ is kind of reciprocal, $z$ root implies $\frac1z$ is also a root$.   All the other things I tried are not conclusive. For example, $a\in[-1,1]$ may be translated as $a=\cos\theta$ for some $\theta\in\mathbb R$, but I don't see how to use this. I tried some rewriting, but nothing seems to work. I tried to work on the modulus of a root, tried the relations roots-coefficients...  I'm quite stuck here. Could you help me please ?  Thanks.
119689
Given $A+B+C=180^{\circ}$, find value of $$\tan A\cdot\tan B+\tan B\cdot\tan C+\tan A\cdot \tan C-\sec A\cdot\sec B\cdot\sec C$$   I know about some basic conditional identities but don't know how to use them here.
63583
My (very simple) question to a friend was how do I prove the following using basic trig principles:  $\tan75^\circ = 2 + \sqrt{3}$  He gave this proof (via a text message!)  $1. \tan75^\circ$   $2. = \tan(60^\circ + (30/2)^\circ)$  $3. = (\tan60^\circ + \tan(30/2)^\circ) / (1 - \tan60^\circ \tan(30/2)^\circ) $  $4. \tan (30/2)^\circ = \dfrac{(1 - \cos30^\circ)}{ \sin30^\circ}$    Can this be explained more succinctly as I'm new to trigonometry and a little lost after (2.) ?    EDIT  Using the answers given I'm almost there:   $\tan75^\circ$    $\tan(45^\circ + 30^\circ)$   $\sin(45^\circ + 30^\circ) / \cos(45^\circ + 30^\circ)$   $(\sin30^\circ.\cos45^\circ + \sin45^\circ.\cos30^\circ) / (\cos30^\circ.\cos45^\circ - \sin45^\circ.\sin30^\circ)$    $\dfrac{(1/2\sqrt{2}) + (3/2\sqrt{2})}{(3/2\sqrt{2}) - (1/2\sqrt{2})}$   $\dfrac{(1 + \sqrt{3})}{(\sqrt{3}) - 1}$   multiply throughout by $(\sqrt{3}) + 1)$   Another alternative approach:   $\tan75^\circ$    $\tan(45^\circ + 30^\circ)$   $\dfrac{\tan45^\circ + \tan30^\circ}{1-\tan45^\circ.\tan30^\circ}$   $\dfrac{1 + 1/\sqrt{3}}{1-1/\sqrt{3}}$   at point 6 in above alternative
7261
I'm wondering if there's a simple way to rewrite this in terms of $k$ and $x$, especially as a polynomial. It seems to me to crop up every so often, especially for $k=2$, when I integrate with trig substitution. But $k=2$ is not so bad, because I can use the double angle formula; it's the prospect of higher values of $k$ that motivates this question.   I think the law of sines may help? Or maybe even De Moivre's theorem, to find the length of the hypotenuse as the length of the angle changes, if we think of the right triangle drawn from $\arcsin(x)$ with side1 = $x$, hypotenuse = $1$, and side2 = $\sqrt{1 - x^2}$ as a complex number, though I'm not sure how that might work.
11349
I've tried substitution, replacing $\sec^2x$ with $\tan^2x + 1$, and parts and I just hit dead ends every time... Do you need knowledge of higher-level calculus to solve this?
11533
I'm tutoring for a college math class and we're doing putnam problems next week and this one stumped me:      Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$.
14883
How do you prove the following trigonometric identity: $$ \sin^2\theta+\cos^2\theta=1$$  I'm curious to know of the different ways of proving this depending on different characterizations of sine and cosine.
82310
To prove  $ \tan(A) + 2 \tan(2A) +4\tan4A + 8 \cot8A =\cot(A) $.  I tried to convert $\tan(4A)$ and $ \tan(8A)$ to $\tan(A$) and tried putting in L.H.S but it becomes a mess. Is there a easy and also intuitive  way to do this?  Thanks
60529
$\sqrt{3}(\cot 70^\circ + 4 \cos 70^\circ ) =$ ?   The answer is $3$.   My progress so far:  \begin{align} \sqrt{3}(\cot 70^\circ+4\cos 70^\circ)&amp;= \sqrt{3}(\tan 20^\circ+4\sin 20^\circ) \\ &amp;= \sqrt{3}(\sin 20^\circ)\left(\frac{1}{\cos 20^\circ}+4\right)\\ &amp;= \sqrt{3}\frac{(\sin 20^\circ)(1+4\cos 20^\circ)}{\cos 20^\circ}\\ &amp;= \sqrt{3}\frac{(\sin 20^\circ+2\sin 40^\circ)}{\cos 20^\circ}\\ \end{align}