sec^-1(-x)=pi-sec^-1 x

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sec-1(-x)=π-sec-1(x).

Let y=sec-1(x), then sec(y)=x, x=1/cos(y).

cos(π-y)=-cos(y), so sec(π-y)=-sec(y)=-x, so π-y=sec-1(-x), or π-sec-1(x)=sec-1(-x).

Therefore sec-1(-x)=π-sec-1(x) QED

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