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