Question

tan47+cot27/tan43+cot63=tan47cot27

Answer 1

Answer: Proved.

Step-by-step explanation:  We are given to prove the following equality

$\dfrac{\tan 47^\circ+\cot 27^\circ}{\tan 43^\circ+\cot 63^\circ}=\tan 47^\circ\cot 27^\circ.$

Let us start as follows -

$L.H.S\\\\=\dfrac{\tan 47^\circ+\cot 27^\circ}{\tan 43^\circ+\cot 63^\circ}\\\\\\=\dfrac{\dfrac{\sin 47^\circ}{\cos 47^\circ}+\dfrac{\cos 27^\circ}{\sin 27^\circ}}{\dfrac{\sin 43^\circ}{\cos 43^\circ}+\dfrac{\cos 63^\circ}{\sin 63^\circ}}\\\\\\=\dfrac{\sin 47^\circ\sin 27^\circ+\cos 47^\circ\cos 27^\circ}{\cos 47^\circ\sin 27^\circ}\times\dfrac{\cos 43^\circ\sin 63^\circ}{\sin 43^\circ\sin 63^\circ+\cos 43^\circ\cos 63^\circ}\\\\\\=\dfrac{\cos(47^\circ-27^\circ)\cos 43^\circ\sin 63^\circ)}{\cos 47^\circ\sin 27^\circ\cos(63^\circ-43^\circ)}\\\\\\=\dfrac{\cos 43^\circ\sin 63^\circ}{\cos 47^\circ\sin 27^\circ}\\\\\\=\dfrac{\cos(90^\circ-47^\circ)\sin(90^\circ-27^\circ)}{\cos 47^\circ\sin 27^\circ}\\\\\\=\dfrac{\sin 47^\circ\cos 27^\circ}{\cos 47^\circ\sin 27^\circ}\\\\\\=\tan 47^\circ\cot 27^\circ\\\\=R.H.S.$

Hence proved.

Answer 2

Answer and explanation:

To prove : $\dfrac{\tan 47^\circ+\cot 27^\circ}{\tan 43^\circ+\cot 63^\circ}=\tan 47^\circ\cot 27^\circ$

Proof :

Taking LHS,

$\dfrac{\tan 47^\circ+\cot 27^\circ}{\tan 43^\circ+\cot 63^\circ}$

$=\dfrac{\dfrac{\sin 47^\circ}{\cos 47^\circ}+\dfrac{\cos 27^\circ}{\sin 27^\circ}}{\dfrac{\sin 43^\circ}{\cos 43^\circ}+\dfrac{\cos 63^\circ}{\sin 63^\circ}}$

$=\dfrac{\sin 47^\circ\sin 27^\circ+\cos 47^\circ\cos 27^\circ}{\cos 47^\circ\sin 27^\circ}\times\dfrac{\cos 43^\circ\sin 63^\circ}{\sin 43^\circ\sin 63^\circ+\cos 43^\circ\cos 63^\circ}$

$=\dfrac{\cos(47^\circ-27^\circ)\cos 43^\circ\sin 63^\circ)}{\cos 47^\circ\sin 27^\circ\cos(63^\circ-43^\circ)}$

$=\dfrac{\cos 43^\circ\sin 63^\circ}{\cos 47^\circ\sin 27^\circ}$

$=\dfrac{\cos(90^\circ-47^\circ)\sin(90^\circ-27^\circ)}{\cos 47^\circ\sin 27^\circ}$

$=\dfrac{\sin 47^\circ\cos 27^\circ}{\cos 47^\circ\sin 27^\circ}$

$=\tan 47^\circ\cot 27^\circ$

= RHS

Hence proved.