Use the fundamental identities to simplify the expression 16

Use the fundamental identities to simplify the expression. 16) 16) cos x (csc x - sec x) - cot x A) cos2x - tan2x B) 0 C) -1 D) 1 coSe 17) sin2 x Cos x sec x A) sec2 x B) cot2 x C) cscx D) csc2 x Use a sum or difference identity to find the exact value 18) sin \"11 18) sin-12 D-4 C) A) B) 2

Solution

16) cosx(cscx -secx) - cotx

= cosx/sinx - cosx*secx -cotx

=cotx -cosx/cosx -cotx

= -cotx

Option C

17) cos^2x/sin^2x + cosxsecx

= cot^2x +cosx/cosx

= cot^2x +1

= csc^2x

Option D

18) sin(11pi/12)

= sin(pi -pi/12) = sinpi/12 { sin(180 -x) =sinx }

= sinpi/12

=sin(pi/3 -pi/4)

Use formula : sin(A-B) = sinAcosB -cosAsinA

sin(pi/3 -pi/4) = sinpi/3cospi/4 -cospi/3sinpi/4

= sqrt(3)/2(1/sqrt2) - (1/2)(1/sqrt2)

= { sqrt(3) - 1}/2sqrt(2)

raionalise denominator:

= sqrt2{ sqrt(3) - 1}/4

= {sqrt(6) - sqrt(2)}/4

Option B