3234363840 Simplify Assume that all radicands are nonnegat s

Simplify. Assume that all radicands are nonnegat sin2 x cos x /cos x 32. Vcos x sin x. sin x 33. V cos sin\' -V/cos\' 34. Vtan x-2 tan x sin x sin x 35. (1 Vsin y) (Vsin y + 1) 36. Vcos ( V 2 cos + V sin cos

Solution

32) sqrt(cos²x sinx)*sqrt(sinx)=cosx*sqrt(sinx)*sqrt(sinx)=cosx*sinx since sqrt(cos²x)=cosxsqrt(sin²x)=sinx

sqrt(cos²x sinx)*sqrt(sinx)=cosx*sinx

34) sqrt(tan²x-2tanx sinx+sin²x)

using the formula a^2-2ab+b^2=(a-b)^2 where a=tanx,b=sinx

sqrt(tan²x-2tanx sinx+sin²x)=sqrt((tanx-sinx)²)=tanx-sinx

sqrt(tan²x-2tanx sinx+sin²x)=tanx-sinx

36) sqrt(cos(theta))*[sqrt(2cos(theta))+sqrt(cos(theta) sin(theta))]

=[sqrt(2cos²(theta))+sqrt(cos²(theta) sin(theta))]

=sqrt(2)*cos(theta)+cos(theta)*sqrt(sin(theta))

=cos(theta)*[sqrt(2)+sqrt(sin(theta))]

sqrt(cos(theta))*[sqrt(2cos(theta))+sqrt(cos(theta) sin(theta))]=cos(theta)*[sqrt(2)+sqrt(sin(theta))]

38) sqrt(cosx/tanx)

Rationalizing the denominator

sqrt(cosx/tanx)*sqrt(tanx/tanx)

=sqrt((cosx tanx)/tan²x)

=sqrt(cosx tanx)/tanx

substitute tanx=sinx/cosx

=sqrt(cosx sinx/cosx)/tanx

=sqrt(sinx)/tanx

sqrt(cosx/tanx) =sqrt(sinx)/tanx

40) sqrt((1-cos)/(1+cos))

Rationalizing the denominator

sqrt((1-cos)/(1+cos))=sqrt((1-cos)/(1+cos))*sqrt(1-cos)/sqrt(1-cos)

=sqrt((1-cos)²/(1-cos²))

use 1-cos²=sin²

=sqrt((1-cos)²/(sin²))

=(1-cos)/sin

sqrt((1-cos)/(1+cos))=(1-cos)/sin