If $$cos4$$⁡θ$$+$$α and $$sin4$$⁡θ$$+$$α are the roots of the equation $$x2+2bx+b=0$$ and $$cos2$$⁡θ$$+$$β,$$sin2$$⁡θ$$+$$β are the roots of the equation $$x2+4x+2=0$$, then values of b are

Question

If $$cos4$$⁡θ$$+$$α and $$sin4$$⁡θ$$+$$α are the roots of the equation $$x2+2bx+b=0$$ and $$cos2$$⁡θ$$+$$β,$$sin2$$⁡θ$$+$$β are the roots of the equation  $$x2+4x+2=0$$, then values of b are

Infinity Learn · Accepted Answer

We have, $$cos2$$⁡θ−$$sin2$$⁡θ$$=$$$$cos$$⁡$$2$$θ⇒ $$cos4$$⁡θ−$$sin4$$⁡θ$$=$$$$cos$$⁡$$2$$θ⇒ $$($$−$$2b)2$$−$$4b=($$−$$4)2$$−$$4$$×$$2(since$$ L.H.S. is difference of roots of first equation and R.H.S. is difference of roots of second $$equation)or$$   $$4b2$$−$$4b=16$$−$$8=8or$$   $$4b2$$−$$4b$$−$$8=0or$$   $$b2$$−b−$$2=0or$$   $$(b+1)(b$$−$$2)=0or$$   $$b=2$$,−$$1$$

If $\cos^{4} θ + α and \sin^{4} θ + α$ are the roots of the equation $x^{2} + 2 bx + b = 0 and \cos^{2} θ + β, \sin^{2} θ + β$ are the roots of the equation $x^{2} + 4 x + 2 = 0$ , then values of b are

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If cos4⁡θ+α and sin4⁡θ+α are the roots of the equation x2+2bx+b=0 and cos2⁡θ+β,sin2⁡θ+β are the roots of the equation x2+4x+2=0, then values of b are

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If $\cos^{4} θ + α and \sin^{4} θ + α$ are the roots of the equation $x^{2} + 2 bx + b = 0 and \cos^{2} θ + β, \sin^{2} θ + β$ are the roots of the equation $x^{2} + 4 x + 2 = 0$ , then values of b are