Question

If A, B, C, D are the angles of a quadrilateral, then (tanA + tanB + tanC + tanD)/(cotA + cotB + cotC + cotD) is equal to

(A) tanA tanB tanC tanD

(B) cotA cotB cotC cotD

(C) tan2A + tan2B + tan2C + tan2D

(D) ∑tan A tanBtanC

Answer 1

Correct option (A)

Explanation:

A + B = 2π – (C + D)

tan(A + B) = – tan(C + D)

(tanA + tanB)/(1 - tanA tanB) = (- tanC - tanD)/(1 - tanC tanD)

tanA – tanA tan C tan D + tan B – tan B tan C tan D = – tanC + tan A tan B tan C – tan D + tan A tan B tan D

tan A + tan B + tan C + tan D = tan A tan C tan D + tan B tan C tan D + tan A tan B tan C + tan A tan B tan D

= tan A tan B tan C tan D (cot A + cot B + cot C + cot D)

∴ = tan A tan B tan C tan D.