Let us consider the equation $$cos4$$⁡$$xa+sin4$$⁡$$xb=1a+b$$,x∈$$0$$,π$$2$$,a,b$$0For$$ the given equation, which of the following is correctThe value of $$sin2x$$ in terms of a and b isThe value of $$sin8$$⁡$$xb3+cos8$$⁡$$xa3$$ is

Question

Let us consider the equation $$cos4$$⁡$$xa+sin4$$⁡$$xb=1a+b$$,x∈$$0$$,π$$2$$,a,b>$$0For$$ the given equation, which of the following is correctThe value of $$sin2x$$ in terms of a and b isThe value of $$sin8$$⁡$$xb3+cos8$$⁡$$xa3$$ is

Infinity Learn · Accepted Answer

We have, $$cos4$$⁡$$xa+sin4$$⁡$$xb=1a+b=cos2$$⁡$$x+sin2$$⁡$$xa+b$$⇒$$cos2$$⁡$$xcos2$$⁡xa−$$1a+b=sin2$$⁡$$x1a+b$$−$$sin2$$⁡xb⇒$$cos2$$⁡$$xbcos2$$⁡x−$$asin2$$⁡$$xa(a+b)a=sin2$$⁡$$xbcos2$$⁡x−$$asin2$$⁡$$xb(a+b)$$⇒$$1$$−$$sin2$$⁡$$xa=sin2$$⁡xb⇒$$Sin2x=ba+b$$ and $$cos2$$⁡$$x=aa+b$$ We have, $$cos4$$⁡$$xa+sin4$$⁡$$xb=1a+b=cos2$$⁡$$x+sin2$$⁡$$xa+b$$⇒$$cos2$$⁡$$xcos2$$⁡xa−$$1a+b=sin2$$⁡$$x1a+b$$−$$sin2$$⁡xb⇒$$cos2$$⁡$$xbcos2$$⁡x−$$asin2$$⁡$$xa(a+b)a=sin2$$⁡$$xbcos2$$⁡x−$$asin2$$⁡$$xb(a+b)$$⇒$$1$$−$$sin2$$⁡$$xa=sin2$$⁡xb⇒$$sin2$$⁡$$x=ba+b$$ and $$cos2$$⁡$$x=aa+b$$  We have, $$cos4$$⁡$$xa+sin4$$⁡$$xb=1a+b=cos2$$⁡$$x+sin2$$⁡$$xa+b$$⇒$$cos2$$⁡$$xcos2$$⁡xa−$$1a+b=sin2$$⁡$$x1a+b$$−$$sin2$$⁡xb⇒$$cos2$$⁡$$xbcos2$$⁡x−$$asin2$$⁡$$xa(a+b)a=sin2$$⁡$$xbcos2$$⁡x−$$asin2$$⁡$$xb(a+b)$$⇒$$1$$−$$sin2$$⁡$$xa=sin2$$⁡xb⇒$$sin2$$⁡$$xb=ba+b$$ and $$cos2$$⁡$$x=aa+b$$⇒$$b3+sin2$$⁡$$x3xa3=b4b3(a+b)4+a3a3(a+b)4$$

Let us consider the equation cos4⁡xa+sin4⁡xb=1a+b,x∈0,π2,a,b>0For the given equation, which of the following is correctThe value of sin2x in terms of a and b isThe value of sin8⁡xb3+cos8⁡xa3 is

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