tan 47°+ cot 27° / tan 43°+ cot 63°= tan 47° cot 27°
Answers
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.