Math, asked by mrinmoyee1974, 6 months ago

tan 47°+ cot 27° / tan 43°+ cot 63°= tan 47° cot 27°​

Answers

Answered by kaushalkavya17
0

Step-by-step explanation:

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.

tan43

+cot63

tan47

+cot27

=tan47

cot27

.

Let us start as follows -

\begin{gathered}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.\end{gathered}

L.H.S

=

tan43

+cot63

tan47

+cot27

=

cos43

sin43

+

sin63

cos63

cos47

sin47

+

sin27

cos27

=

cos47

sin27

sin47

sin27

+cos47

cos27

×

sin43

sin63

+cos43

cos63

cos43

sin63

=

cos47

sin27

cos(63

−43

)

cos(47

−27

)cos43

sin63

)

=

cos47

sin27

cos43

sin63

=

cos47

sin27

cos(90

−47

)sin(90

−27

)

=

cos47

sin27

sin47

cos27

=tan47

cot27

=R.H.S.

Hence proved.

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