Question

question 26 (mandatory) (1 point)
which expression is equivalent to ( 1 - 2 sin ^ { 2 } \theta ) ?
a) ( 1 - 2 cos ^ { 2 } \theta )
b) ( cos ^ { 2 } \theta )
c) ( sin ^ { 2 } \theta )
d) ( 2 cos ^ { 2 } \theta - 1 )

Explanation:

Step1: Recall the double - angle formula for cosine

The double - angle formula for cosine is \(\cos2\theta=\cos^{2}\theta-\sin^{2}\theta\). Also, since \(\sin^{2}\theta+\cos^{2}\theta = 1\), we can express \(\cos^{2}\theta=1 - \sin^{2}\theta\).
Substitute \(\cos^{2}\theta\) into the double - angle formula: \(\cos2\theta=(1 - \sin^{2}\theta)-\sin^{2}\theta=1-2\sin^{2}\theta\).
We can also express \(\sin^{2}\theta = 1-\cos^{2}\theta\). Substitute \(\sin^{2}\theta\) into the double - angle formula \(\cos2\theta=\cos^{2}\theta-(1 - \cos^{2}\theta)=2\cos^{2}\theta-1\).

Answer:

D. \(2\cos^{2}\theta - 1\)