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Your question is insufficiently precise, but I'll try to answer anyway.

"Sine squared theta" usually means "the value of the sine of theta, quantity squared".

"Sine theta squared" usually means "the value of the sine of the quantity theta*theta".

The two are not at all the same.

Q: Is sin squared theta and sin theta squared the same?

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It is 2*sin(theta)*sin(theta) because that is how multiplication is defined!

2 sin^2 theta = 1/4 sin^2 theta = 1/8 sin theta = sqrt(1/8) theta = arcsin(sqrt(1/8))

It's 1/2 of sin(2 theta) .

If r-squared = theta then r = Â±sqrt(theta)

If there is a plus in between, that would be equal to 1, as a result of the Pythagorean Theorem. Otherwise, you can convert this into other forms with some of the trigonometric identities for multiplication, but you won't really get it into a simpler form.

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You can use the Pythagorean identity to solve this:(sin theta) squared + (cos theta) squared = 1.

It is 2*sin(theta)*sin(theta) because that is how multiplication is defined!

1

Until an "equals" sign shows up somewhere in the expression, there's nothing to prove.

2 sin^2 theta = 1/4 sin^2 theta = 1/8 sin theta = sqrt(1/8) theta = arcsin(sqrt(1/8))

COS squared Theta + SIN squared Theta = 1; where Theta is the angles measurement in degrees.

Cosine squared theta = 1 + Sine squared theta

sin(0)=0 and sin(very large number) is approximately equal to that same very large number.

Tan^2

It's 1/2 of sin(2 theta) .

If r-squared = theta then r = Â±sqrt(theta)

It is 1.