Question 1209558
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<U>Part &nbsp;(a)</U>



Let's consider part &nbsp;(a) &nbsp;first, &nbsp;with equation 


    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;7^(sin^2(x)) - 7^(cos^2(x)) = 8.



Due to &nbsp;OBVIOUS &nbsp;reasons, &nbsp;this equation has no real solutions.


Indeed, &nbsp;sin^2(x) &nbsp;has the maximum value of &nbsp;1;  &nbsp;hence,  &nbsp;7^(sin^2(x)) &nbsp;has maximum value of 7.


Next, &nbsp;from &nbsp;7^(sin^2(x)), &nbsp;the equation subtracts &nbsp;7^(cos^2(x)), &nbsp;which is non-negative
real number, &nbsp;so the left side of the equation &nbsp;CAN &nbsp;NOT &nbsp;be greater than &nbsp;7.


    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;A fortiori, &nbsp;it can not be &nbsp;8.


At this point, &nbsp;my solution is complete, and the 


<U>ANSWER</U> &nbsp;&nbsp;is: &nbsp;&nbsp;this given equation has no solution/solutions in real numbers.



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The solution by the &nbsp;@CPhill to this problem has a &nbsp;FATAL &nbsp;ERROR:


it incorrectly factors &nbsp;&nbsp;y^2 - 8y - 7 = 0  &nbsp;as  &nbsp;&nbsp;(y-7)*(y+1) = 0,


which leads him to further wrong conclusions.



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This example/case tells me that @CPhill 


&nbsp;&nbsp;&nbsp;&nbsp;- does not look at his solutions;


&nbsp;&nbsp;&nbsp;&nbsp;- does not read his solutions;


&nbsp;&nbsp;&nbsp;&nbsp;- does not check them;


&nbsp;&nbsp;&nbsp;&nbsp;- does not think about them;


&nbsp;&nbsp;&nbsp;&nbsp;- does not care about the correctness of his solutions,


and, &nbsp;in general, &nbsp;does not understand, &nbsp;what he is doing and what he posts 
to this forum, in particular, and to the outer world, &nbsp;in general.



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    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<U>Part &nbsp;(c)</U>  



In the post by &nbsp;@CPhill, &nbsp;the solution for part &nbsp;(c) &nbsp;is &nbsp;INCORRECT.


The correct solution is given at this forum under this link


https://www.algebra.com/algebra/homework/playground/test.faq.question.1209563.html



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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In future, do not pack more than one problem per post.


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