Question

tan6*tan42*tan66*tan78=1

Answer 1

There is a formula, tan(x)*tan(60+x)*tan(60-x) = tan(3x) Using that formula, tan 6 * tan66 * tan54 * tan18 * tan78 * tan42 tan6 * tan42 * tan66 * tan78 = --------------------------------------------- tan 18 * tan 54 tan 6*tan(60+6)*tan(60-6)* tan18*tan(60+18)*tan(60-18) tan6 * tan42 * tan66 * tan78 = -------------------------------------------------------- tan 18 * tan 54 tan(3*6) tan(3*18) tan 18 * tan 54 tan6*tan42*tan66*tan78 = -------------------- = ---------------- = 1 proved tan 18 * tan 54 tan 18 * tan 54 OR tan 6 * tan66 * tan54 * tan18 * tan78 * tan42 tan6 * tan42 * tan66 * tan78 = --------------------------------------------- tan 18 * tan 54 sin6*sin(60+6)*sin(60-6)*sin18*sin(60+18)*sin(60-18) tan6*tan42*tan66*tan78 = --------------------------------------------------------------------- cos6*cos(60+6)*cos(60-6)*cos18*cos(60+18)*cos(60-18)(tan 18*tan 54) tan 18 * tan 54 sin6 *(sin²60 -sin²6)*sin18*(sin²60 -sin²18) tan6*tan42*tan66*tan78 = ------------------------------------------------------------- cos6*(cos²60-sin²6)*cos18*(cos²(60-sin²18)(tan 18*tan 54) sin6*(3/4-sin²6)*sin18*(3/4-sin²18) tan6*tan42*tan66*tan78 = --------------------------------------------------- cos6*(1/4-sin²6)*cos18*(1/4-sin²18)(tan18*tan 54) sin6*(3 -4sin²6)*sin18*(3-sin²18) tan6*tan42*tan66*tan78 = ------------------------------------------------- cos6*(1-sin²6)*cos18*(1-sin²18)(tan18*tan 54) (3sin6-4sin²6)*(3sin18-4sin²18) tan6*tan42*tan66*tan78 = ------------------------------------------------- (4cos³6-3cos6)*(4cos³18-3cos18)(tan 18*tan54) sin(3*6)*sin(3*18) tan6*tan42*tan66*tan78 = -------------------------------- cos(3*6)*cos(3*18)(tan18*tan 54) sin18*sin54 tan6*tan42*tan66*tan78 = -------------------------- cos18*cos54(tan18*tan54) tan18*tan54 tan6*tan42*tan66*tan78 = --------------- = 1 proved tan18*tan54