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

Question

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.

$\frac{cos^2A+cos^2A}{2\sin A\cos A}=\frac{2\cos^2A}{2\sin A \cos A}=\cot A$ — Saurabh, Jul 26 '12 at 17:56
There's also a nice way of seeing this. Take a look at this answer by robjohn. — Ian Mateus, Jul 26 '12 at 18:24

score 5 · Answer 1 · answered Jul 26 '12 at 18:14

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

double angle formulas
multiply numerator and denominator by $\csc^2(A)$
$\cot^2(A)+1=\csc^2(A)$
cancel $2\cot(A)$ in numerator and denominator

score 5 · Answer 2 · edited May 09 '13 at 19:16

5

$$\frac{1+\cos(2A)}{\sin(2A)} = \frac{2\cos^2 A}{2\sin A \cdot \cos A} = \frac{\cos A}{\sin A} = \cot A$$

edited May 09 '13 at 19:16

davidlowryduda

91,687

answered May 09 '13 at 18:40

iostream007

4,529

1

I have rolled back to TMM's edit, since it is exceptionally clear and easy to read. – davidlowryduda May 09 '13 at 19:17
1

why is this downvoted – Sidd Singal May 09 '13 at 19:19
thanks @mixedmath. should I write the answer if someone already give – iostream007 May 09 '13 at 19:42

score 3 · Answer 3 · answered Jul 26 '12 at 17:58

3

Just write $\cos^2 A+\cos^2 A=2\cos^2 A$ (a quantity added to itself is twice the quantity). Then write $2\cos^2 A=2\cos A\cdot\cos A$ and cancel a $2\cos A$ term in the numerator with the $2\cos A$ term in the denominator.

answered Jul 26 '12 at 17:58

David Mitra

74,748

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

3 Answers3