Answer
Let's rewrite the expression on the left in terms of the ratios of sides:
sin $A$ = $\frac{opposite}{hypotenuse}$
tan $A$ = $\frac{opposite}{adjacent}$
Let's plug in what we know into the equation we are given:
$\frac{1}{\frac{a^2}{c^2}} - \frac{1}{\frac{a^2}{b^2}} = 1$
Rewrite the equation to simplify:
$\frac{c^2}{a^2} - \frac{b^2}{a^2} = 1$
Rewrite the fractions as one fraction:
$\frac{c^2 - b^2}{a^2} = 1$
We now look at the Pythagorean theorem, which relates the sides to the hypotenuse. The Pythagorean theorem is given by the following formula:
$a^2 + b^2 = c^2$
If we were to rewrite the Pythagorean theorem in terms of $a^2$, then we would have:
$a^2 = c^2 - b^2$
We can replace $c^2 - b^2$ with $a^2$:
$\frac{a^2}{a^2} = 1$
Simplify the fraction by dividing both the numerator and denominator by their greatest common factor:
$1 = 1$
The identity is true.
Work Step by Step
--
