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Sign of Trigonometric Ratios in Different Quadrants

If A,B,C,D ar...

Question

If $A, B, C, D$ are the angles of a cyclic quadrilateral, then the value of $sec A + sec B + sec C + sec D$ is

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Solution

As $A B C D$ is cyclic quadrilateral
$∴ A + C = π$ and $B + D = π$
$\Rightarrow A = π - C$ and $B = π - D$

Now,
$sec A + sec B + sec C + sec D = sec (π - C) + sec (π - D) + sec C + sec D = - sec C - sec D + sec C + sec D = 0$

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