Question

Let ABCD (in order) is a cyclic quadrilateral. Which of the following is/are true?
A
sec B = sec D
B
Cot A + Cot C = 0
C
cos ECA = cos ecc
D
tan B + tan B = 0​

Answer 1

$\underline{\textsf{Given:}}$

$\textsf{ABCD is a cyclic quadrilateral}$

$\underline{\textsf{To find:}}$

$\textsf{Most appropriate answer in the given alternatives}$

$\underline{\textsf{Solution:}}$

$\textsf{We know that,}$

$\textsf{OPPOSITE ANGLES OF A CYCLIC QUADRILATERAL}$

$\textsf{ARE SUPPLEMENTARY}$

$\implies\mathsf{A+C=180^{\circ}\;\;\&\;\;B+D=180^{\circ}}$

$\textsf{Consider,}$

$\mathsf{A+C=180^{\circ}}$

$\mathsf{A=180^{\circ}-C}$

$\implies\mathsf{cotA=cot(180^{\circ}-C)}$

$\implies\mathsf{cotA=-cotC}$

$\implies\boxed{\mathsf{cotA+cotC=0}}$

$\textsf{Consider,}$

$\mathsf{B+C=180^{\circ}}$

$\mathsf{B=180^{\circ}-D}$

$\implies\mathsf{tanB=tan(180^{\circ}-D)}$

$\implies\mathsf{tanB=-tanD}$

$\implies\boxed{\mathsf{tanB+tanD=0}}$

$\underline{\textsf{Answer:}}$

$\textsf{Options (B) and (D) are correct}$