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# How do you verify sec^4x-sec^2x=tan^4x+tan^2x?

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56
Feb 22, 2016

Verify ${\sec}^{4} x - {\sec}^{2} x = {\tan}^{4} x + {\tan}^{2} x$

#### Explanation:

Left side $\to {\sec}^{4} x - {\sec}^{2} x$

$= \frac{1}{{\cos}^{4} x} - \frac{1}{{\cos}^{2} x}$

$= \frac{1 - {\cos}^{2} x}{{\cos}^{4} x}$

$= {\sin}^{2} \frac{x}{{\cos}^{4} x}$

$= {\tan}^{2} x \left(\frac{1}{{\cos}^{2} x}\right)$

Apply the trig identity: $\frac{1}{{\cos}^{2} x} = \left(1 + {\tan}^{2} x\right)$, we get:

Left side $\to {\tan}^{2} x \left(1 + {\tan}^{2} x\right) = {\tan}^{4} x + {\tan}^{2} x .$

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