1. ## Trigonometry Question

2. Hello, raj1985!

Is there a typo? .The statement isn't true.

Prove: .If $\displaystyle 3\sec A-7\tan A\:=\:3$, then: .$\displaystyle 3\tan A-7\sec A\:=\:{\color{red}-}7$ . ??

We have: .$\displaystyle 3\sec A \:=\:3 + 7\tan A$

Square both sides: .$\displaystyle 9\sec^2\!A \:=\:9 + 42\tan A + 49\tan^2\!A$

. . . . . . . . . . .$\displaystyle 9(\tan^2\!A + 1) \:=\:9 + 42\tan A + 49\tan^2\!A$

. . . . . . . . . . . $\displaystyle 9\tan^2\!A + 9 \:=\:9 + 42\tan A + 49\tan^2\!A$

. . . . . . $\displaystyle 40\tan^2\!A - 42\tan A \:=\:0$

. . . . . . $\displaystyle 2\tan\theta(20\tan A - 21) \:=\:0$

We have: .$\displaystyle 2\tan\theta \:=\:0\quad\Rightarrow\quad \theta \,=\,0$ .
for which the statement is true.

And: .$\displaystyle 20\tan A - 21 \:=\:0\quad\Rightarrow\quad \tan A \:=\:\frac{21}{20}$

. . . $\displaystyle \sec^2\!A \:=\:\tan^2\!A + 1 \:=\:\left(\frac{21}{20}\right)^2 + 1 \:=\:\frac{841}{400} \quad\Rightarrow\quad \sec A \:=\:\frac{29}{20}$

Then: .$\displaystyle 3\tan A - 7\sec A \;=\;3\left(\frac{21}{20}\right) - 7\left(\frac{29}{20}\right) \;=\;\frac{63}{20} - \frac{203}{20} \;=\;\frac{\text{-}140}{20} \;=\;{\color{red}-7}$