Question

Sederhanakan identitas trigonometri di bawah ini
[tex]\displaystyle \frac{\cot x}{\cot x-\cot 3x}+\frac{\tan x}{\tan x-\tan 3x}[/tex]

Answer 1

Verified answer

Penjelasan dengan langkah-langkah:

Identitas trigonometri

• cot x = [tex]\sf\frac{1}{\tan~x}[/tex]
• cot 3x = [tex]\sf\frac{1}{\tan~3x}[/tex]

[tex]\begin{aligned} \sf \frac{ \cot(x) }{ \cot(x) - \cot(3x) } + \frac{ \tan(x) }{ \tan(x) - \tan(3x) } & = \sf \frac{ \cot(x) }{ \frac{1}{ \tan(x) } - \frac{1}{ \tan(3x) } } + \frac{ \tan(x) }{ \tan(x) - \tan(3x) } \\ & = \sf \frac{ \cot(x) }{ \frac{ \tan(3x) - \tan(x) }{ \tan(x) \tan(3x) } } + \frac{ \tan(x) }{ - 1( \tan(3x) - \tan(x) )} \\ & = \sf \frac{ \frac{1}{ \cancel{\tan(x)} } }{\frac{ \tan(3x) - \tan(x) }{ \cancel{\tan(x)} \tan(3x) }} - \frac{ \tan(x) }{ \tan(3x) - \tan(x) } \\& = \sf \frac{ \tan(3x) }{ \tan(3x) - \tan(x) } - \frac{ \tan(x) }{ \tan(3x) - \tan(x) } \\ & = \sf \frac{ \tan(3x) - \tan(x) }{ \tan(3x) - \tan(x) } \\& =\boxed{ 1} \sf \end{aligned}[/tex]