Question

Buktikan identitas hiperbolik​

Answer 1

Jawab:

Terbukti semua

Penjelasan dengan langkah-langkah:

Nomor 1.

cosh 2x = cosh (x + x)

= cosh x cosh x + sinh x sinh x

= cosh² x + sinh² x

Nomor 2.

(sinh x + cosh x) = (-i sin ix + cos ix)

Jika dikuadratkan

(sinh x + cosh x)² = (-i sin ix + cos ix)²

= -sin² ix - 2i sin ix cos ix + cos² ix

= cos² ix - sin² ix - i sin 2ix

= cos 2ix - i sin 2ix

= cosh 2x + sinh 2x

Jadi

(sinh x + cosh x)² = sinh 2x + cosh 2x

Jika 2 diganti dengan n maka

(sinh x + cosh x)ⁿ = sinh nx + cosh nx

Nomor 3

[tex]\displaystyle \tanh (x-y)=\frac{\sinh (x-y)}{\cosh (x-y)}\\=\frac{\sinh x\sinh y-\cosh x\cosh y}{\cosh x\cosh y-\sinh x\sinh y}\\=\frac{\frac{\sinh x\sinh y-\cosh x\cosh y}{\cosh x\cosh y}}{\frac{\cosh x\cosh y-\sinh x\sinh y}{\cosh x\cosh y}}\\=\frac{\tanh x-\tanh y}{1-\tanh x\tanh y}[/tex]

Nomor 4

[tex]\displaystyle \cosh 2x=\cosh^2x+\sinh^2 x\\\cosh 2x=\cosh^2 x+\cosh^2 x-1\\\cosh x+1=2\cosh ^2\left ( \frac{x}{2} \right )\\\cosh^2 \left ( \frac{x}{2} \right )=\frac{\cosh x+1}{2}[/tex]

Nomor 5

[tex]\displaystyle \tanh x=\sqrt{\frac{\cosh x-1}{\cosh x+1}}\\=\sqrt{\frac{\cosh x-1}{\cosh x+1}~\frac{\cosh x-1}{\cosh x-1}}\\=\sqrt{\frac{(\cosh x-1)^2}{\cosh^2 x-1}}\\=\sqrt{\frac{(\cosh x-1)^2}{\sinh^2 x}}\\=\frac{\cosh x-1}{\sinh x}\\=\frac{\cosh x-1}{\sinh x}~\frac{\cosh x+1}{\cosh x+1}\\=\frac{\cosh^2 x-1}{\sinh x(\cosh x+1)}\\=\frac{\sinh x}{1+\cosh x}[/tex]