ill in the blank to complete the trigonometric formula. 8sin(u)+cos(u)_(times )=(8sin(u))/(1+cos(u))

To fill in the blank and complete the trigonometric formula, we need to find the value that should be placed in the blank to make the equation true.<br /><br />Given equation: $8\sin(u) + \cos(u)_{\times} = \frac{8\sin(u)}{1 + \cos(u)}$<br /><br />To solve this, we can use the trigonometric identity $\sin^2(u) + \cos^2(u) = 1$.<br /><br />Let's rewrite the equation as:<br />$8\sin(u) + \cos(u)_{\times} = \frac{8\sin(u)}{1 + \cos(u)}$<br /><br />Now, let's multiply both sides of the equation by $(1 + \cos(u))$ to eliminate the denominator on the right side:<br />$(8\sin(u) + \cos(u)_{\times})(1 + \cos(u)) = 8\sin(u)$<br /><br />Expanding the left side, we get:<br />$8\sin(u) + 8\sin(u)\cos(u) + \cos(u) + \cos^2(u)_{\times} = 8\sin(u)$<br /><br />Now, let's simplify the equation by combining like terms:<br />$8\sin(u) + 8\sin(u)\cos(u) + \cos(u) + \cos^2(u)_{\times} - 8\sin(u) = 0$<br /><br />Simplifying further, we get:<br />$8\sin(u)\cos(u) + \cos(u) + \cos^2(u)_{\times} = 0$<br /><br />Now, let's factor out $\cos(u)$ from the first two terms:<br />$\cos(u)(8\sin(u) + 1) + \cos^2(u)_{\times} = 0$<br /><br />To make the equation true, the blank should be filled with $-8\sin(u)$.<br /><br />Therefore, the completed trigonometric formula is:<br />$8\sin(u) + \cos(u)_{\times} = \frac{8\sin(u)}{1 + \cos(u)}$<br /><br />So, the value that should be placed in the blank is $-8\sin(u)$.