Linked Questions

0
votes
3answers
5k views

Show $\tan(x)+\tan(y)+\tan(z) = \tan(x) \tan(y) \tan(z)$ [duplicate]

I am not able to show that: If $x+y+z=\pi$, show that $\tan(x) + \tan(y) + \tan(z) = \tan(x) \tan(y) \tan(z)$.
3
votes
1answer
4k views

Prove that : $\tan 40 + \tan 60 + \tan 80 = \tan 40 \cdot \tan 60 \cdot \tan 80$ [duplicate]

I started from Left hand side as 3^1/2 + tan 2(20) +tan 4(20). But that brought me a lot of terms to solve which ends (9 tan 20 - 48 tan^3 20 -50 tan^5 20 - 16 tan^7 20 + tan^9 20)/(1- 7 tan^2 20 + 7 ...
3
votes
2answers
117 views

How to prove the the addition of tangent is the same as the multiplication? [duplicate]

If A,B,C are angles of a triangle show that: $$\tan A+ \tan B+\tan C = \tan A \tan B \tan C $$ I've tried this many times but I cannot seem to prove it, can someone show me how to solve this problem?...
62
votes
5answers
19k views

Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$?

Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this ...
4
votes
4answers
2k views

If $\alpha = \frac{2\pi}{7}$ then the find the value of $\tan\alpha .\tan2\alpha +\tan2\alpha \tan4\alpha +\tan4\alpha \tan\alpha.$

If $\alpha = \frac{2\pi}{7}$ then the find the value of $\tan\alpha .\tan2\alpha +\tan2\alpha \tan4\alpha +\tan4\alpha \tan\alpha$ My 1st approach : $\tan(\alpha +2\alpha +4\alpha) = \frac{\tan\...
2
votes
3answers
1k views

How to show that $\log (\frac{2a}{1-a^2}+\frac{2b}{1-b^2}+\frac{2c}{1-c^2})= \log\frac{2a}{1-a^2}+ \log \frac{2b}{1-b^2}+ \log \frac{2c}{1-c^2}$

If $\log (a +b +c) =\log a+\log b+\log c$ then show that $$\log \left(\frac{2a}{1-a^2}+\frac{2b}{1-b^2}+\frac{2c}{1-c^2}\right)= \log\frac{2a}{1-a^2}+ \log \frac{2b}{1-b^2}+ \log \frac{2c}{1-c^2}$$ ...
4
votes
4answers
2k views

Problem in the solution of a trigonometric equation $\tan\theta + \tan 2\theta+\tan 3\theta=\tan\theta\tan2\theta\tan3\theta$

I needed to solve the following equation: $$\tan\theta + \tan 2\theta+\tan 3\theta=\tan\theta\tan2\theta\tan3\theta$$ Now, the steps that I followed were as follows. Transform the LHS first: $$\...
4
votes
7answers
709 views

How to proceed from $\cot(x)\cot(2x)-\cot(2x)\cot(3x)-\cot(3x)\cot(x) = 1$

To prove: $\cot(x)\cot(2x)-\cot(2x)\cot(3x)-\cot(3x)\cot(x) = 1$ My attempt at the solution: \begin{gather}\frac{\cos(x)\cos(2x)}{\sin(x)\sin(2x)}-\frac{\cos(2x)\cos(3x)}{\sin(2x)\sin(3x)}-\frac{\cos(...
1
vote
2answers
790 views

If $\tan^3A+\tan^3B+\tan^3C=3\tan(A)\tan(B)\tan(C)$, prove triangle ABC is equilateral triangle

If $\tan^3A+\tan^3B+\tan^3C=3\tan(A)\tan(B)\tan(C)$, prove triangle ABC is equilateral triangle Now i remember a identity which was like if $a+b+c=0$,then $a^3+b^3+c^3=3abc$. So i have $\sum_{}^{} \...
6
votes
3answers
222 views

Question related to tan in a ratio in a triangle

If in a triangle $\tan A:\tan B:\tan C = 1:2:3$ then, what are the ratio of the sides $a,b,c $?
0
votes
3answers
880 views

Find the third angle of the triangle

Question: If two angles of a triangle $ABC$ are $\arctan 2$ and $\arctan 3$, what is the third angle? My attempt: Let the third angle of the triangle $ABC$ be $x$. $\therefore$ $\arctan 2+\arctan 3+ ...
4
votes
3answers
389 views

$\tan{A} \cdot \tan{B} \cdot \tan{C}=9$, find $\tan^2{A}+\tan^2{B}+ \tan^2{C}$

In $\triangle{ABC}$, $$\tan{A}\cdot \tan{B}\cdot \tan{C}=9$$ $$\tan^2{A}+\tan^2{B}+ \tan^2{C}=\lambda$$ then, $\lambda$ lies in the interval?
0
votes
3answers
1k views

Find the smallest angle in the triangle

The point H is the orthocenter of the triangle ABC and the point C is the centroid of the triangle ABH. In that case the smallest angle of the triangle ABC is: (60), (30), (45), ($\angle ACB$)? This ...
4
votes
2answers
156 views

How prove this $\alpha+\beta+\gamma=n\pi$

let $\theta\in R$,and $\alpha\neq\beta\neq\gamma$ and such $$\dfrac{\cos{(\alpha+\theta)}}{\sin^3{\alpha}}=\dfrac{\cos{(\beta+\theta)}}{\sin^3{\beta}}=\dfrac{\cos{(\gamma+\theta)}}{\sin^3{\gamma}}$$ ...
2
votes
2answers
341 views

How to prove $\tan3°\tan63°\tan69°=\tan15°$?

Prove $\tan3°\tan63°\tan69°=\tan15°$ And assuming we don't know that $\tan15^{\circ}$ part, how to just evaluate $\tan3^{\circ} tan63^{\circ} tan69^{\circ}$?

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