Print Page - Trigonometry simplify question with three variables

Spoiler

$ \begin{align*}\frac{\log_{\frac{a}{b}}(x)\log_{\frac{a}{c}}(x)}{(\log_a(x))^2}+\frac{\log_{\frac{b}{c}}(x)\log_{\frac{b}{a}}(x)}{(\log_b(x))^2}+\frac{\log_{\frac{c}{a}}(x)\log_{\frac{c}{b}}(x)}{(\log_c(x))^2} &= \frac{\frac{(\ln(x))^2}{\ln \left(\frac{a}{b}\right) \dot \ \ln \left(\frac{a}{c}\right)}}{\frac{(\ln(x))^2}{(\ln(a))^2}} + \frac{\frac{(\ln(x))^2}{\ln \left(\frac{b}{c}\right) \dot \ \ln \left(\frac{b}{a}\right)}}{\frac{(\ln(x))^2}{(\ln(b))^2}} + \frac{\frac{(\ln(x))^2}{\ln \left(\frac{c}{a}\right) \dot \ \ln \left(\frac{c}{b}\right)}}{\frac{(\ln(x))^2}{(\ln(c))^2}} \\&= \frac{(\ln (a))^2}{\ln \left(\frac{a}{b}\right) \dot \ \ln \left(\frac{a}{c}\right)} + \frac{(\ln (b))^2}{\ln \left(\frac{b}{c}\right) \dot \ \ln \left(\frac{b}{a}\right)} + \frac{(\ln (c))^2}{\ln \left(\frac{c}{a}\right) \dot \ \ln \left(\frac{c}{b}\right)} \\&= \frac{(\ln (a))^2}{(\ln (a) - \ln (b))(\ln(a) - \ln (c))} + \frac{(\ln (b))^2}{(\ln (b) - \ln (c))(\ln(b) - \ln (a))} + \frac{(\ln (c))^2}{(\ln (c) - \ln (a))(\ln(c) - \ln (b))} \\&= \frac{(\ln (a))^2(\ln (c)- \ln (b))}{(\ln (a) - \ln (b))(\ln(a) - \ln (c))(\ln (c) - \ln (b))} + \frac{(\ln (b))^2(\ln(a) - \ln (c))}{(\ln (c) - \ln (b))(\ln(a) - \ln (b))(\ln(a) - \ln (c))} + \frac{(\ln (c))^2(\ln (b) - \ln (a))}{(\ln (c) - \ln (a))(\ln(c) - \ln (b))(\ln (b) - \ln (a))} \\&= 1 \end{align*} $

ATAR Notes: Forum

HSC Stuff => HSC Maths Stuff => HSC Subjects + Help => HSC Mathematics Advanced => Topic started by: Arfalt on March 24, 2021, 10:25:38 pm