verify the identity.\n\n cos left( x + \frac { pi } { 4 } \right) = \frac { sqrt { 2 } } { 2 } ( cos x - sin…

verify the identity.\n\n cos left( x + \frac { pi } { 4 } \right) = \frac { sqrt { 2 } } { 2 } ( cos x - sin x ) \n\nwrite the left side in terms of a sum or difference formula for sine or cosine.\n\n cos left( x + \frac { pi } { 4 } \right) =

verify the identity.\n\n cos left( x + \frac { pi } { 4 } \right) = \frac { sqrt { 2 } } { 2 } ( cos x - sin x ) \n\nwrite the left side in terms of a sum or difference formula for sine or cosine.\n\n cos left( x + \frac { pi } { 4 } \right) =

Answer

Explanation:

Step1: Apply the cosine sum formula

The cosine sum formula is $\cos(A + B)=\cos A\cos B-\sin A\sin B$. Here $A = x$ and $B=\frac{\pi}{4}$. So, $\cos\left(x+\frac{\pi}{4}\right)=\cos x\cos\frac{\pi}{4}-\sin x\sin\frac{\pi}{4}$.

Step2: Substitute the values of $\cos\frac{\pi}{4}$ and $\sin\frac{\pi}{4}$

We know that $\cos\frac{\pi}{4}=\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$. Substituting these values, we get $\cos x\times\frac{\sqrt{2}}{2}-\sin x\times\frac{\sqrt{2}}{2}$.

Step3: Factor out the common term

Factor out $\frac{\sqrt{2}}{2}$ from the expression: $\frac{\sqrt{2}}{2}(\cos x-\sin x)$.

Answer:

$\frac{\sqrt{2}}{2}(\cos x - \sin x)$