1) Prove the following identity.
( 3 - 6 cos² x ) / ( sin x - cos x ) = 3 ( sin x + cos x )
2) Prove that
(a) cot x + tan x = cosec x sec x
(b) cosec² x + sec² x = cosec² x sec² x
Hence, deduce that (cot x + tan x )² = cosec² x + sec² x
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Answer:
1)
( 3 - 6 cos² x )
= 3 - 3cos² x -3cos² x
= 3( sin² x - cos² x )
= 3 (sin x + cos x) (sin x - cos x)
Hence, ( 3 - 6 cos² x ) / ( sin x - cos x ) = 3 ( sin x + cos x )
2)
a)
cos x/sin x + sin x/cos x
= ( cos²x + sin²x ) / (sin x cos x)
= 1 / (sin x cos x)
= cosec x sec x
b)
1 / sin² x + 1 / cos² x
= ( cos² x + sin² x ) / (sin² x cos² x)
= 1 / (sin² x cos² x)
= cosec² x sec² x
LHS
= (cot x + tan x )²
= cosec² x sec² x
= cosec² x + sec² x (from part b)
