Question

write the expression as the sine, cosine, or tangent of a double - angle. then find the exact value of the expression. cos²180° - sin²180° which expression is cos²180° - sin²180° equal to? select the correct choice below and fill in the answer box within your choice. (simplify your answer. type your answer in degrees. use integers or decimals for any numbers in the expression.) a. cos²180° - sin²180° = sin ° b. cos²180° - sin²180° = cos 360° c. cos²180° - sin²180° = tan ° the exact value of the expression is. (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize the denominator.)

Explanation:

Step1: Use double - angle formula

The double - angle formula for cosine is \(\cos2\alpha=\cos^{2}\alpha-\sin^{2}\alpha\). Here \(\alpha = 180^{\circ}\), so \(\cos^{2}180^{\circ}-\sin^{2}180^{\circ}=\cos(2\times180^{\circ})=\cos360^{\circ}\)

Step2: Calculate the value of \(\cos360^{\circ}\)

We know that the cosine function has a period of \(360^{\circ}\), i.e., \(\cos(x + 360^{\circ}n)=\cos x\) for any integer \(n\). So \(\cos360^{\circ}=\cos(0^{\circ}+ 360^{\circ})=\cos0^{\circ}\)
Since \(\cos0^{\circ}=1\)

Answer:

The exact value of the expression \(\cos^{2}180^{\circ}-\sin^{2}180^{\circ}\) is \(1\)