**1(a)**

given #tan2theta=-20/21#

Let #tantheta =x #

So using the identity #tan2theta=(2tantheta)/(1-tan^2theta)# we get

#-20/21=(2x)/(1-x^2)#

#=>-10/21=x/(1-x^2)#

#=>10x^2-21x-10=0#

#=>10x^2-25x+4x-10=0#

#=>5x(2x-5)+2(2x-5)=0#

#=>(2x-5)(5x+2)=0#

So #x=5/2 and x=-2/5#

Hence #tantheta=5/2 and tantheta=-2/5#

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**1(b)**

given #tan2theta=-36/77#

Let #tantheta =x #

So using the identity #tan2theta=(2tantheta)/(1-tan^2theta)# we get

#36/77=(2x)/(1-x^2)#

#=>18/77=x/(1-x^2)#

#=>18x^2+77x-18=0#

#=>18x^2+81x-4x-18=0#

#=>9x(2x+9)-2(2x+9)=0#

#=>(2x+9)(9x-2)=0#

So #x=-9/2 and x=2/9#

Hence #tantheta=-9/2 and tantheta=2/9#

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**2(a)**

#cos2theta+5costheta=2#

#=>2cos^2theta-1+5costheta-2=0#

#=>2cos^2theta+5costheta-3=0#

#=>2cos^2theta+6costheta-costheta-3=0#

#=>(2costheta-1)(costheta+3)=0#

So #costheta=-3->"not possible"#

when

#(2costheta-1)=0#

#=>costheta=1/2=cos60^@=cos(360-60)^@=cos300^@#

Hence #theta=60^@ and theta=300^@#

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**2(b)**

#2sec2x-cot2x=tan2x#

#=>2sec2x=tan2x+cot2x#

#=>2sec2x=(sin2x)/(cos2x)+(cos2x)/(sin2x)#

#=>2sec2x=(sin^2 2x+cos^2 2x)/(sin2xcos2x)=csc2xsec2x#

#=>2sec2x-csc2xsec2x=0#

#=>sec2x(2-csc2x)=0#

#sec2x=0-> "not possible"#

when #(2-csc2x)=0#

#sin2x=1/2=sin30^@=sin(180-30)=sin150^@#

Hence #x= 15^@ and x= 75^@#

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**2(c)**

#2tan(theta/2)+3tantheta=0#

#=>2tan(theta/2)+(6tan(theta/2))/(1-tan^2(theta/2))=0#

#=>2tan(theta/2)(1+3/(1-tan^2(theta/2)))=0#

#=>2tan(theta/2)(4-tan^2(theta/2))=0#

when

#tan(theta/2)=0=tan(o^@)=tan180^@#

Hence #theta = 0^@ and theta = 360^@#

#(4-tan^2(theta/2))=0#

#=>tan(theta/2)=+-2#

when

#=>tan(theta/2)=+2=tan63.43^@#

#=>theta =126.86^@#

when

#=>tan(theta/2)=-2=-tan63.43^@=tan(180-63.43)#

#=>tan(theta/2)=tan116.57^@#

#=>theta =233.14^@#