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How do you prove $1 + sin² x = 1/sec² x$ ?

1 Answer

Jul 5, 2015

The given equation is not true (except for values of $x$ equal to integer multiples of $pi$ ).

Explanation:

$sec(x) = 1/cos(x)$

The maximum value of $abs(cos(x))$ is 1

$rArr$ the minimum value of $abs(sec(x))$ is 1
and
the maximum value of $1/(sec^2(x))$ is 1

$sin^2(x)$ has a minimum value of 0
$1+sin^2(x)$ has a minimum value of 1

The only time $1+sin^2(x) = 1/(sec^2(x))$
is when
$color(white)("XXXX")$ $sec^2(x) = 1$ and $sin^2(x) = 1$

These conditions are only true for $x=0$ and any other value of $x=kpi$ (with $kepsilonZZ$ ).

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