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

verify the identity. \n\\( \\cos \\left( x + \\frac { \\pi } { 4 } \\right) = \\frac { \\sqrt { 2 } } { 2 } ( \\cos x - \\sin x ) \\)\nwrite the left side in terms of a sum or difference formula for sine or cosine.\n\\( \\cos \\left( x + \\frac { \\pi } { 4 } \\right) = \\cos x \\cos \\frac { \\pi } { 4 } - \\sin \\frac { \\pi } { 4 } \\sin x \\)\nrewrite the expression by evaluating the sine and cosine of \\( \\frac { \\pi } { 4 } \\).\n(rationalize all denominators. simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

verify the identity. \n\\( \\cos \\left( x + \\frac { \\pi } { 4 } \\right) = \\frac { \\sqrt { 2 } } { 2 } ( \\cos x - \\sin x ) \\)\nwrite the left side in terms of a sum or difference formula for sine or cosine.\n\\( \\cos \\left( x + \\frac { \\pi } { 4 } \\right) = \\cos x \\cos \\frac { \\pi } { 4 } - \\sin \\frac { \\pi } { 4 } \\sin x \\)\nrewrite the expression by evaluating the sine and cosine of \\( \\frac { \\pi } { 4 } \\).\n(rationalize all denominators. simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Answer

Explanation:

Step1: Use the cosine sum formula

The formula for (\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: Evaluate (\cos\frac{\pi}{4}) and (\sin\frac{\pi}{4})

We know that (\cos\frac{\pi}{4}=\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}). Substitute these values into the above - expression: (\cos\left(x + \frac{\pi}{4}\right)=\cos x\times\frac{\sqrt{2}}{2}-\sin x\times\frac{\sqrt{2}}{2}).

Step3: Factor out (\frac{\sqrt{2}}{2})

Factor out the common factor (\frac{\sqrt{2}}{2}) from the right - hand side. We get (\cos\left(x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\cos x-\sin x)).

Answer:

The identity (\cos\left(x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\cos x - \sin x)) is verified.