Question

Prove (cosecø-secø)(cotø-tanø)=(cosecø+secø)(secøcosecø-2)​

Answer 1

Step-by-step explanation:

taking LHS=

(cosec¢-sec¢)(cot¢-tan¢)

=(1/sin¢-1/cos¢)(cos¢/sin¢-sin¢/cos¢)

=(cos¢-sin¢/sin¢cos¢)(cos^2¢-sin^2¢/sin¢cos¢)

=(cos¢-sin¢)(cos¢+sin¢)(cos¢-sin¢)

sin^2¢cos^2¢

=(cos¢+sin¢)(cos¢-sin¢)^2

sin^2¢cos^2¢

=(cos¢+sin¢)(cos^2¢+sin^2¢-2sin¢cos¢)

sin^2¢cos^2¢

=(cos¢ +sin¢)(1-2sin¢cos¢)

sin^2¢cos^¢

Then taking RHS=

( cosec¢+sec¢)(sec¢cosec¢-2)

=(1/sin¢+1/cos¢)(1/sin¢cos¢-2)

=(cos¢+sin¢/sin¢cos¢)(1-2sin¢cos¢/sin¢cos¢)

=(cos¢+sin¢)(1-2sin¢cos¢)

sin^2¢cos^¢

LHS=RHS

hence proved