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Question Number 211961 by Nadirhashim last updated on 25/Sep/24
   if   7^(sin^(2 ) x) + 7^(cos^2 x) = 8 find x
$$\:\:\:\boldsymbol{{if}}\:\:\:\mathrm{7}^{\boldsymbol{{sin}}^{\mathrm{2}\:} \boldsymbol{{x}}} +\:\mathrm{7}^{\boldsymbol{{cos}}^{\mathrm{2}} \boldsymbol{{x}}} =\:\mathrm{8}\:\boldsymbol{{find}}\:\boldsymbol{{x}} \\ $$
Answered by efronzo1 last updated on 25/Sep/24
  7^(sin^2 x)  = a ⇒7^(cos^2 x) = 7^(1−sin^2 x) = (7/a)   ⇒a +(7/a) = 8   ⇒a^2 −8a+7=0    ⇒a=1 or a=7    { ((7^(sin^2 x  ) =1⇒sin^2 x=0)),((7^(sin^2 x)  =7⇒sin^2 x=1)) :}
$$\:\:\mathrm{7}^{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}} \:=\:\mathrm{a}\:\Rightarrow\mathrm{7}^{\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}} =\:\mathrm{7}^{\mathrm{1}−\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}} =\:\frac{\mathrm{7}}{\mathrm{a}} \\ $$$$\:\Rightarrow\mathrm{a}\:+\frac{\mathrm{7}}{\mathrm{a}}\:=\:\mathrm{8} \\ $$$$\:\Rightarrow\mathrm{a}^{\mathrm{2}} −\mathrm{8a}+\mathrm{7}=\mathrm{0} \\ $$$$\:\:\Rightarrow\mathrm{a}=\mathrm{1}\:\mathrm{or}\:\mathrm{a}=\mathrm{7} \\ $$$$\:\begin{cases}{\mathrm{7}^{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}\:\:} =\mathrm{1}\Rightarrow\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}=\mathrm{0}}\\{\mathrm{7}^{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}} \:=\mathrm{7}\Rightarrow\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}=\mathrm{1}}\end{cases} \\ $$
Answered by BaliramKumar last updated on 26/Sep/24
x = 0, (π/2)
$${x}\:=\:\mathrm{0},\:\frac{\pi}{\mathrm{2}} \\ $$

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