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Trigonometric Identities

Show that non...

Question

Show that none of the following is an identity :

(i) $c o s^{2} θ + c o s θ = 1$ (ii) $s i n^{2} θ + s i n θ = 2$ (iii) $t a n^{2} θ + s i n θ = c o s^{2} θ$

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Solution

Identity is satisfied by all values of the angle, so here we will prove that these equations are satisfied by only some values of the angle.
$1 right parenthesis space cos squared theta space plus space cos theta space equals space 1 cos squared theta space plus space cos theta space minus 1 space equals space 0 t h i s space i s space a space q u a d r a t i c space e q u a t i o n space i n space cos theta s o space b y space t h e space s o l u t i o n space o f space q u a d r a t i c space e q u a t i o n s space w h i c h space i s space fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction w e space g e t comma space fraction numerator negative 1 plus-or-minus square root of 1 minus 4. left parenthesis negative 1 right parenthesis. left parenthesis 1 right parenthesis end root over denominator 2. left parenthesis 1 right parenthesis end fraction space equals space fraction numerator negative 1 plus-or-minus square root of 5 over denominator 2 end fraction space s o space cos theta space equals space fraction numerator negative 1 plus square root of 5 over denominator 2 end fraction space o r space cos theta space equals space fraction numerator negative 1 minus square root of 5 over denominator 2 end fraction sin c e space t h e r e space i s space o n l y space o n e space v a l u e space o f space theta space f o r space o n e space v a l u e space o f space cos theta space h e n c e space t h i s space e q u a t i o n space i s space n o t space a n space i d e n t i t y$
$2 right parenthesis space sin squared theta space plus space sin theta space equals space 2 sin squared theta space plus space sin theta space minus 2 space equals space 0 t h i s space i s space a space q u a d r a t i c space e q u a t i o n space i n space sin theta s o space b y space t h e space s o l u t i o n space o f space q u a d r a t i c space e q u a t i o n s space w h i c h space i s space fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction w e space g e t comma space fraction numerator negative 1 plus-or-minus square root of 1 minus 4. left parenthesis negative 2 right parenthesis. left parenthesis 1 right parenthesis end root over denominator 2. left parenthesis 1 right parenthesis end fraction space equals space fraction numerator negative 1 plus-or-minus square root of 10 over denominator 2 end fraction space s o space sin theta space equals space fraction numerator negative 1 plus square root of 10 over denominator 2 end fraction almost equal to 1.081 space o r space sin theta space equals space fraction numerator negative 1 minus square root of 10 over denominator 2 end fraction almost equal to negative 2.08 sin c e space t h e s e space v a l u e s space a r e space g r e a t e r space t h a n space 1 space o r space l e s s space t h a n space minus 1 space h e n c e space t h e space e q u a t i o n space i s space n o t space s a t i s f i e d space b y space a n y space v a l u e s space o f space theta a s space sin theta space c a n space o n l y space b e space b e t w e e n space minus 1 space a n d space 1 space s o space t h e space e q u a t i o n space i s space n o t space a n space i d e n t i t y.$
$left parenthesis 3 right parenthesis space tan squared theta space plus space sin theta space equals space cos squared theta fraction numerator sin squared theta over denominator cos squared theta end fraction space plus space sin theta space equals space cos squared theta fraction numerator sin squared theta over denominator 1 minus sin squared theta end fraction space plus space sin theta space equals space 1 minus sin squared theta sin squared theta space plus space sin theta left parenthesis 1 minus sin squared theta right parenthesis space equals space left parenthesis 1 minus sin squared theta right parenthesis squared sin squared theta space plus space sin theta minus sin cubed theta space equals space 1 minus 2 sin squared theta plus sin to the power of 4 theta sin to the power of 4 theta space plus space sin cubed theta space minus 3 sin squared theta space minus sin theta space plus 1 space equals space 0 sin c e space t h i s space i s space a space b i q u a d r a t i c space e q u a t i o n space i n space sin theta comma space h e n c e space t h i s space c a n space o n l y space h a v e space m a x i m u m space 4 space v a l u e s space o f space theta space t h a t space s a t i s f y space t h e space e q u a t i o n h e n c e space t h i s space i s space n o t space a n space i d e n t i t y.$

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