(cos(frac (1pi )/(2)-1)-cos(2pi -1))(tan(pi +t))=

To solve this expression, we need to simplify the numerator and denominator separately.<br /><br />In the numerator, we have two cosine functions. Let's simplify them one by one.<br /><br />The first cosine function is $\cos(\frac{\pi}{2}-1)$. We can use the co-function identity for cosine and sine to simplify this expression. The co-function identity states that $\cos(\frac{\pi}{2}-x) = \sin(x)$. Applying this identity, we get:<br /><br />$\cos(\frac{\pi}{2}-1) = \sin(1)$<br /><br />The second cosine function is $\cos(2\pi-1)$. We can use the periodic property of cosine to simplify this expression. The periodic property states that $\cos(x+2\pi) = \cos(x)$. Applying this property, we get:<br /><br />$\cos(2\pi-1) = \cos(-1) = \cos(1)$<br /><br />Now, let's simplify the numerator by substituting the simplified expressions for the cosine functions:<br /><br />$\cos(\frac{\pi}{2}-1) - \cos(2\pi-1) = \sin(1) - \cos(1)$<br /><br />In the denominator, we have the tangent function. We can use the periodic property of tangent to simplify this expression. The periodic property states that $\tan(x+\pi) = \tan(x)$. Applying this property, we get:<br /><br />$\tan(\pi+t) = \tan(t)$<br /><br />Now, let's simplify the entire expression by substituting the simplified expressions for the numerator and denominator:<br /><br />$\frac{\sin(1) - \cos(1)}{\tan(t)}$<br /><br />This is the simplified form of the given expression.