How do you simplify the expression ${cos}^{2} A ({sec}^{2} A - 1)$ $cos^2A(sec^2A-1)$ ?

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How do you simplify the expression ${cos}^{2} A ({sec}^{2} A - 1)$ $cos^2A(sec^2A-1)$ ?

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Answer 1 · 2016-10-01T08:20:28

${cos}^{2} A ({sec}^{2} A - 1) = {sin}^{2} A$ $cos^2 A (sec^2 A - 1) = sin^2 A$

with exclusion $A \neq \frac{π}{2} + n π$ $A != pi/2 + npi$ for integer values of $n$ $n$ .

Explanation:

Note that:

$sec A = \frac{1}{cos A}$ $sec A = 1/(cos A)$

${sin}^{2} A + {cos}^{2} A = 1$ $sin^2 A + cos^2 A = 1$

So we find:

${cos}^{2} A ({sec}^{2} A - 1) = \frac{{cos}^{2} A}{{cos}^{2} A} - {cos}^{2} A$ $cos^2 A (sec^2 A - 1) = (cos^2 A)/(cos^2 A) - cos^2 A$

${cos}^{2} A ({sec}^{2} A - 1) = 1 - {cos}^{2} A$ $color(white)(cos^2 A (sec^2 A - 1)) = 1 - cos^2 A$

${cos}^{2} A ({sec}^{2} A - 1) = {sin}^{2} A$ $color(white)(cos^2 A (sec^2 A - 1)) = sin^2 A$

Note that this identity does not hold for $A = \frac{π}{2} + n π$ $A = pi/2 + npi$ , when $sec A$ $sec A$ is undefined, resulting in the left hand side being undefined but the right hand side defined.

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How do you simplify the expression ${cos}^{2} A ({sec}^{2} A - 1)$ $cos^2A(sec^2A-1)$ ?

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How do you simplify the expression ${cos}^{2} A ({sec}^{2} A - 1)$ $cos^2A(sec^2A-1)$ ?