\( (\sin (-x)+\cos (360-x))(\cos (90-x)+\frac{\sin (x)}{\tan (x)})simplifywithoutusingacadlculator \)

Let's simplify the expression step by step using trigonometric identities. The given expression is: \[ (\sin (-x) + \cos (360^\circ - x)) \left( \cos (90^\circ - x) + \frac{\sin x}{\tan x} \right) \] ### Step 1: Simplify Each Term Inside the Parentheses 1. **Simplify \(\sin (-x)\):** \[ \sin (-x) = -\sin x \quad \text{(since sine is an odd function)} \] 2. **Simplify \(\cos (360^\circ - x)\):** \[ \cos (360^\circ - x) = \cos x \quad \text{(since cosine is periodic with period } 360^\circ \text{)} \] So, \[ \cos (360^\circ - x) = \cos x \] 3. **Simplify \(\cos (90^\circ - x)\):** \[ \cos (90^\circ - x) = \sin x \quad \text{(using the co-function identity)} \] 4. **Simplify \(\frac{\sin x}{\tan x}\):** Recall that \(\tan x = \frac{\sin x}{\cos x}\), so: \[ \frac{\sin x}{\tan x} = \frac{\sin x}{\frac{\sin x}{\cos x}} = \cos x \] ### Step 2: Substitute the Simplified Terms Back into the Expression Now, substitute the simplified terms back into the original expression: \[ (\sin (-x) + \cos (360^\circ - x)) \left( \cos (90^\circ - x) + \frac{\sin x}{\tan x} \right) = (-\sin x + \cos x)(\sin x + \cos x) \] ### Step 3: Expand the Expression Notice that the expression \((-\sin x + \cos x)(\sin x + \cos x)\) is of the form \((a - b)(a + b)\), which simplifies to \(a^2 - b^2\). Applying this: \[ (-\sin x + \cos x)(\sin x + \cos x) = \cos^2 x - \sin^2 x \] ### Step 4: Apply the Double Angle Identity Recall the double angle identity for cosine: \[ \cos 2x = \cos^2 x - \sin^2 x \] Therefore, \[ \cos^2 x - \sin^2 x = \cos 2x \] ### Final Simplified Expression \[ \boxed{\cos 2x} \]