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1h
comment Difficult inverse tangent identity
Its not given equality, but a proposition to be verified. If you take $\theta=\arcsin(x)$ by the definition of Principal value,$$-\dfrac\pi2\le x\le\dfrac\pi2\implies\cos\theta\ge0$$ and consequently $$|\cos\theta|=+\cos\theta$$
2h
comment Difficult inverse tangent identity
For the sake of completeness, you should explain why $$\sqrt{1-\sin^2\theta}=|\cos\theta|$$ reduces to $$+\cos\theta$$
2h
comment reflection of the point $(1,0)$
@Mathxx, Please find the edited version
2h
comment For which $n$ does $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ imply $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}$
@LinusS. The derivation is even easier & natural if we write $$\dfrac1a+\dfrac1b=\dfrac1{a+b+c}-\dfrac1c$$ to get $$(a+b)(c^2+bc+ca+ab)=0$$ and $$c^2+bc+ca+ab=(c+a)(c+b)$$
18h
comment Find an acute angle $\gamma$ such that $\sin \gamma + \cos \gamma= \sqrt{2}$
See en.wikibooks.org/wiki/Trigonometry/… or en.wikipedia.org/wiki/Tangent_half-angle_substitution
19h
comment Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$
possible duplicate of Highest power of a prime $p$ dividing $N!$
20h
comment Finding extreme values of a variable on an intersection of a sphere and a plane
@Lozansky, en.wikipedia.org/wiki/Lagrange_multiplier $f_x$ is the first order partial derivative wrt $x$
20h
comment Finding extreme values of a variable on an intersection of a sphere and a plane
@Lozansky, Using Lagrange multipliers $$f(x,y,z)=z+a(x^2+y^2+z^2-1)+b(x+2y+2z)$$ Then find $f_x,f_y,f_z$
20h
comment Solving $7[x]+23\{x\}=191$
@Integrator, Nice to see your response
20h
comment Is it possible to have $a^2 + b^2 = c^2 + 1$ for $a$, $b$, $c$ being coprime integers?
Wish I know the mistake here. One can easily check with example like $$a=26,b=21\implies d=4,c=5$$
20h
comment Finding extreme values of a variable on an intersection of a sphere and a plane
@Lozansky, Arranged it as a Quadratic Equation in $y$
22h
comment The maclaurin series $ f(x) =\frac {x^3} {2+ x^2}$
$$=x^3/2\left(1+x^2/2\right)^{-1}$$
22h
comment $a \exp(iwt)+b \exp(-iwt)=0 \Rightarrow a = b = 0$
Use en.wikipedia.org/wiki/Euler%27s_formula
22h
comment Finding extreme values of a variable on an intersection of a sphere and a plane
This discriminant is very useful is determining extreme values like in math.stackexchange.com/questions/1303772/…
1d
comment Obtaining $\sum_{n=1}^{\infty} a^n \cos{(n\theta)} = \frac{a \cos{\theta}-a^2}{1-2a\cos{\theta}+a^2}$
@Snowflake, See Article$\#76$ of archive.org/details/treatiseonplanet00hobs
1d
comment Extract sum of coefficients in a binomial expression
@nam, have you understood my post correctly?
1d
comment Tell if a sum is convergent $\sum\limits_{n=1}^\infty \frac{2}{n(n+1)}$
See en.wikipedia.org/wiki/Telescoping_series
1d
comment Extract sum of coefficients in a binomial expression
@nam, Observe that $k_r=0$ for $3n+1\le r\le6n$ So, $$\sum_{r=1}^{6n}k_rc_r=\sum_{r=1}^{3n}k_rc_r=$$
1d
comment Extract sum of coefficients in a binomial expression
@nam, Welcome! Are you sure that it is not $$c_0+c_1+\cdots+c_{6n}?$$ What's the source of the question ?
1d
comment How to integrate $\int dx \frac{1}{\cosh^2 x +a^2}$
Multiply with sech$^2x$