To prove this identity, we can first use the even and odd properties of cosine and sine functions:
cos(-x) = cos(x) (evensin(-x) = -sin(x) (odd)
Therefore, the given expression can be written as:
cos^2(-x) + sin(-x) = cos^2(x) - sin(x)
Next, we can use the Pythagorean identity cos^2(x) + sin^2(x) = 1 to simplify the expression further:
cos^2(x) - sin(x= cos^2(x) - (1 - cos^2(x)) [Using Pythagorean identity sin^2(x)=1-cos^2(x)= cos^2(x) - 1 + cos^2(x= 2cos^2(x) - 1
Therefore, the given expression cos^2(-x) + sin(-x) simplifies to 2cos^2(x) - 1.
Now, we can use the Pythagorean identity again to simplify the right side of the equation:
2 - sin^2(x= 2 - (1 - cos^2(x)= 2 - 1 + cos^2(x= 1 + cos^2(x)
So, the right side simplifies to 1 + cos^2(x).
Since we have shown that cos^2(-x) + sin(-x) simplifies to 2cos^2(x) - 1, we can rewrite the original expression as:
2cos^2(x) - 1 = 1 + cos^2(x)
Simplifying this expression further, we get:
2cos^2(x) - 1 = 1 + cos^2(xcos^2(x) = 2
Therefore, the identity cos^2(-x) + sin(-x) = 2 - sin^2(x) is proved to be true.
To prove this identity, we can first use the even and odd properties of cosine and sine functions:
cos(-x) = cos(x) (even
sin(-x) = -sin(x) (odd)
Therefore, the given expression can be written as:
cos^2(-x) + sin(-x) = cos^2(x) - sin(x)
Next, we can use the Pythagorean identity cos^2(x) + sin^2(x) = 1 to simplify the expression further:
cos^2(x) - sin(x
= cos^2(x) - (1 - cos^2(x)) [Using Pythagorean identity sin^2(x)=1-cos^2(x)
= cos^2(x) - 1 + cos^2(x
= 2cos^2(x) - 1
Therefore, the given expression cos^2(-x) + sin(-x) simplifies to 2cos^2(x) - 1.
Now, we can use the Pythagorean identity again to simplify the right side of the equation:
2 - sin^2(x
= 2 - (1 - cos^2(x)
= 2 - 1 + cos^2(x
= 1 + cos^2(x)
So, the right side simplifies to 1 + cos^2(x).
Since we have shown that cos^2(-x) + sin(-x) simplifies to 2cos^2(x) - 1, we can rewrite the original expression as:
2cos^2(x) - 1 = 1 + cos^2(x)
Simplifying this expression further, we get:
2cos^2(x) - 1 = 1 + cos^2(x
cos^2(x) = 2
Therefore, the identity cos^2(-x) + sin(-x) = 2 - sin^2(x) is proved to be true.