Buktikan Identitas Trigonometri Dari, Tan A-1/Tan A+1=Sec A-Cosec A/Sec A+Cosec A

Buktikan identitas trigonometri dari, tan A-1/tan A+1=sec A-cosec A/sec A+cosec A

 

Jawab dan Penjelasan dengan langkah-langkah:

\large\text{$\begin{aligned}&\frac{\tan A-1}{\tan A+1}=\frac{\ \dfrac{\sin A}{\cos A}-1\ }{\dfrac{\sin A}{\cos A}+1}\end{aligned}$}

\large\text{$\begin{aligned}&\quad<br>ormalsize\begin{array}{|ll}\sin A=\frac{1}{\mathrm{cosec}\,A}\\\\\frac{1}{\cos A}=\sec A\\\\\frac{\sin A}{\cos A}=\sin A\cdot\frac{1}{\cos A}\\\\\qquad\ \;=\frac{1}{\mathrm{cosec}\,A}\cdot\sec A\\\\\qquad\ \;=\frac{\sec A}{\mathrm{cosec}\,A}\end{array}\\\\&{=\ }\frac{\ \dfrac{\sec A}{\mathrm{cosec}\,A}-1\ }{\dfrac{\sec A}{\mathrm{cosec}\,A}+1}\end{aligned}$}

\large\text{$\begin{aligned}&{=\ }\frac{\ \dfrac{\sec A-\mathrm{cosec}\,A}{\cancel{\mathrm{cosec}\,A}}\ }{\dfrac{\sec A+\mathrm{cosec}\,A}{\cancel{\mathrm{cosec}\,A}}}\\\\&{=\ }\frac{\sec A-\mathrm{cosec}\,A}{\sec A+\mathrm{cosec}\,A}\end{aligned}$}

Dengan demikian, terbukti benar bahwa:

\large\text{$\begin{aligned}&\boxed{\ \frac{\tan A-1}{\tan A+1}=\frac{\sec A-\mathrm{cosec}\,A}{\sec A+\mathrm{cosec}\,A}\ }\end{aligned}$}


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