Question

Show that none of the following is an identity:
` (i) cos^(2) theta + cos theta =1 " " (ii) sin^(2) theta + sin theta =2 " " (iii) tan^(2) theta + sin theta = cos^(2) theta `

Answer 1

`(i) cos^(2)30^(@)+cos 30^(@)=((sqrt(3))/(2))^(2)+(sqrt(3))/(2)=(3)/(4)+(sqrt(3))/(2)=(3+2sqrt(3))/(4) ne 1`.
(ii) `sin^(2)30^(@)+sin30^(@)=((1)/(2))^(2)+(1)/(2)=(1)/(4)+(1)/(2)=(3)/(4) ne 2.`
(iii) ` LHS=tan^(2)45^(@)+sin45^(@)=(1)^(2)+(1)/(sqrt(2))=1+(1)/(sqrt(2))=(sqrt(2)+1)/(sqrt(2))`.
`RHS=cos^(2)45^(@)=((1)/(sqrt(2)))^(2)=(1)/(2).`
`therefore LHS ne RHS.`