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