rewrite the expression as a simplified expression containing one term.\n\\( \\cos ( \\frac { \\pi } { 2 } +…

rewrite the expression as a simplified expression containing one term.\n\\( \\cos ( \\frac { \\pi } { 2 } + \\alpha ) \\cos ( \\frac { \\pi } { 2 } - \\alpha ) - \\sin ( \\frac { \\pi } { 2 } + \\alpha ) \\sin ( \\frac { \\pi } { 2 } - \\alpha ) = \\square \\)\n(type an integer, a simplified fraction, or a simplified expression.)

rewrite the expression as a simplified expression containing one term.\n\\( \\cos ( \\frac { \\pi } { 2 } + \\alpha ) \\cos ( \\frac { \\pi } { 2 } - \\alpha ) - \\sin ( \\frac { \\pi } { 2 } + \\alpha ) \\sin ( \\frac { \\pi } { 2 } - \\alpha ) = \\square \\)\n(type an integer, a simplified fraction, or a simplified expression.)

Answer

Explanation:

Step1: Use co - function identities

Recall the co - function identities: (\cos(A)=\sin\left(\frac{\pi}{2}-A\right)) and (\sin(A)=\cos\left(\frac{\pi}{2}-A\right)). So, (\cos\left(\frac{\pi}{2}+\alpha\right)=\sin(-\alpha)=-\sin\alpha), (\cos\left(\frac{\pi}{2}-\alpha\right)=\sin\alpha), (\sin\left(\frac{\pi}{2}+\alpha\right)=\cos\alpha), (\sin\left(\frac{\pi}{2}-\alpha\right)=\cos\alpha).

Step2: Substitute into the original expression

Substitute these identities into the expression (\cos\left(\frac{\pi}{2}+\alpha\right)\cos\left(\frac{\pi}{2}-\alpha\right)-\sin\left(\frac{\pi}{2}+\alpha\right)\sin\left(\frac{\pi}{2}-\alpha\right)). We get ((-\sin\alpha)\times\sin\alpha-\cos\alpha\times\cos\alpha).

Step3: Simplify the expression

Using the formula (a\times a = a^{2}), the expression becomes (-\sin^{2}\alpha-\cos^{2}\alpha). Since (\sin^{2}\alpha+\cos^{2}\alpha = 1), then (-(\sin^{2}\alpha+\cos^{2}\alpha)=- 1).

Answer:

(-1)