\( \sin (90-x) \cdot \cos \left(360^{\circ} x\right)+\tan x \cos (-x) \sin (150- \) Shat that \( \sin (360-x) \cdot \tan ^{2} x \) \( \sin \left(x-180^{\circ}\right) \cos (90+x)=\frac{1}{\cos x} \)

Let's break down the problem step by step. We have the following expressions to analyze: 1. \( \sin (90-x) \cdot \cos (360^{\circ} x) + \tan x \cos (-x) \sin (150) - \sin (360-x) \cdot \tan^{2} x \) 2. \( \sin (x-180^{\circ}) \cos (90+x) = \frac{1}{\cos x} \) ### Step 1: Simplifying the First Expression We will simplify the first expression step by step. 1. **Using Trigonometric Identities:** - \( \sin(90 - x) = \cos x \) - \( \cos(360^{\circ} x) = \cos(0) = 1 \) (since cosine is periodic with a period of \(360^{\circ}\)) - \( \tan x = \frac{\sin x}{\cos x} \) - \( \cos(-x) = \cos x \) - \( \sin(150^{\circ}) = \frac{1}{2} \) - \( \sin(360 - x) = -\sin x \) (since sine is periodic with a period of \(360^{\circ}\)) Now substituting these identities into the expression: \[ \cos x \cdot 1 + \frac{\sin x}{\cos x} \cdot \cos x \cdot \frac{1}{2} - (-\sin x) \cdot \tan^{2} x \] This simplifies to: \[ \cos x + \frac{1}{2} \sin x - \sin x \cdot \tan^{2} x \] 2. **Substituting \( \tan^{2} x = \frac{\sin^{2} x}{\cos^{2} x} \):** \[ \cos x + \frac{1}{2} \sin x - \sin x \cdot \frac{\sin^{2} x}{\cos^{2} x} \] This can be further simplified, but let's keep it in this form for now. ### Step 2: Simplifying the Second Expression Now let's simplify the second expression: \[ \sin (x-180^{\circ}) \cos (90+x) = \frac{1}{\cos x} \] 1. **Using Trigonometric Identities:** - \( \sin(x - 180^{\circ}) = -\sin x \) - \( \cos(90 + x) = -\sin x \) Substituting these identities gives: \[ -\sin x \cdot (-\sin x) = \frac{1}{\cos x} \] This simplifies to: \[ \sin^{2} x = \frac{1}{\cos x} \] ### Step 3: Final Formulation Now we have two simplified expressions: 1. From the first expression: \[ \cos x + \frac{1}{2} \sin x - \sin x \cdot \frac{\sin^{2} x}{\cos^{2} x} \] 2. From the second expression: \[ \sin^{2} x = \frac{1}{\cos x} \] ### Conclusion The first expression can be further simplified or evaluated based on specific values of \(x\), while the second expression gives a relationship between \(\sin^{2} x\) and \(\cos x\). If you have specific values for \(x\) or further instructions on what to do with these expressions, please let me know!