If \( \operatorname{Sin} \theta=45 \) \( \frac{\operatorname{Tan}^{2} \theta+\operatorname{Sin}^{2} \theta}{\cos ^{2} \theta+\operatorname{Tan} \theta} \) \( \begin{array}{ll}\text { (A) } 1.2 & \text { (D) } 36 \\ \text { (B) } 42 & \text { (E) } 135 \\ \text { (C) } 121 & \end{array} \)
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It looks like there is a misunderstanding in the question because the sine function, \( \sin \theta \), cannot equal 45, as its values range only between -1 and 1. This suggests that perhaps the question meant \( \sin \theta = \frac{1}{\sqrt{2}} \) (which is approximately 0.7071) for \( \theta = 45^\circ \). If you meant to evaluate the expression for \( \theta = 45^\circ \), then we can proceed. Given: \[ \sin 45^\circ = \frac{1}{\sqrt{2}}, \quad \cos 45^\circ = \frac{1}{\sqrt{2}}, \quad \tan 45^\circ = 1 \] Substituting these values into the expression, we have: \[ \frac{\tan^2(45^\circ) + \sin^2(45^\circ)}{\cos^2(45^\circ) + \tan(45^\circ)} \] \[ = \frac{1^2 + \left(\frac{1}{\sqrt{2}}\right)^2}{\left(\frac{1}{\sqrt{2}}\right)^2 + 1} \] \[ = \frac{1 + \frac{1}{2}}{\frac{1}{2} + 1} \] \[ = \frac{\frac{3}{2}}{\frac{3}{2}} \] \[ = 1 \] So the answer is \( 1 \). However, since 1 is not listed as one of the options, please check if there was a mistake in the given values or options.
