Answer:
see derivation below
Explanation:
Show that:
( sec(t) - cosec(t) ) ( 1 + tan(t) + cot(t) ) =
sec(t) tan(t) - cosec(t) cot(t)
Some trigonometric definitions used:
tan(t) = sin(t)/cos(t)
cot(t) = cos(t)/sin(t)
sec(t) = 1/cos(t)
csc(t) = 1/sin(t)
some trigonometric identities used:
sin^2(t) + cos^2(t) = 1 ......................(1)
rewrite left-hand side in terms of sine and cosine
(1/cos(t) - 1/sin(t) ) ( 1 + sin(t)/cos(t) + cos(t)/sin(t) )
Simplify using common denominator sin(t)cos(t)
= ( (sin(t) - cos(t))/(sin(t)*cos(t)) ) * ( ( sin(t)cos(t) + sin^2(t) + cos^2(t)) / ( sin(t)cos(t) ) )
= ( sin(t) -cos(t) ) * (1 + sin(t)cos(t) ) / ( sin^2(t) cos^2(t) ) ...... using (1)
Expand by multiplication
= ( sin(t) -cos(t) + sin^2(t)cos(t) - sin(t)cos^2(t) ) / ( sin^2(t) cos^2(t) )
Rearrange by factoring out sin(t) and cos(t) in numerator
= ( sin(t) (1-cos^2(t) - cos(t)(1-sin^2(t) ) / ( sin^2(t) cos^2(t) )
= ( sin^3(t) - cos^3(t) ) /( sin^2(t) cos^2(t) ) .........................using (1)
Cancel common factors
= sin(t)/(cos^2(t)) - cos(t)/(sin^2(t))
Rewrite using trigonometric definitions
= sec(t)tan(t) - csc(t)cot(t) as in Right-Hand Side