[Math] Prove $\frac{1+\cos{(2A)}}{\sin{(2A)}}=\cot{A}$

trigonometry

I am sorry to ask so many of these questions in such as short time span.

But how would I prove this following trigonometric identity.
$$
\frac{1+\cos(2A)}{\sin(2A)}=\cot A
$$
My work thus far is
$$
\frac{1+\cos^2A-\sin^2A}{2\sin A\cos A}
$$
I know $1-\sin^2A=\cos^2A$

So I do
$$
\frac{\cos^2A+\cos^2A}{2\sin A\cos A}
$$
I know not what I do next.

Best Answer

$$ \begin{align} \frac{1+\cos(2A)}{\sin(2A)} &=\frac{1+\cos^2(A)-\sin^2(A)}{2\sin(A)\cos(A)}\tag{1}\\ &=\frac{\csc^2(A)+\cot^2(A)-1}{2\,\cot(A)}\tag{2}\\ &=\frac{2\,\cot^2(A)}{2\,\cot(A)}\tag{3}\\[4pt] &=\cot(A)\tag{4} \end{align} $$

  1. double angle formulas

  2. multiply numerator and denominator by $\csc^2(A)$

  3. $\cot^2(A)+1=\csc^2(A)$

  4. cancel $2\cot(A)$ in numerator and denominator

Related Question