State whether the following statements are correct:
\sin ^{2}\theta + \cos ^{2}\theta = 1
\sin ^{2}\theta + \cos ^{2}\theta = 2
Find the exact value of the following:
Simplify the following expressions:
\left(\cos \theta - 1\right) \left(\cos \theta + 1\right)
\left(\cos \theta - \sin \theta\right)^{2}
\cos \theta \sin ^{2}\left(\theta\right) - \cos \theta
\left(3 - \cos x\right)^{2} + \sin ^{2}\left(x\right)
\dfrac{\sin ^{2}\left(\theta\right)}{1 - \sin ^{2}\left(\theta\right)}
\dfrac{1 - \sin ^{2}\left(\theta\right)}{\sin ^{2}\left(\theta\right) + \cos ^{2}\left(\theta\right)}
If x = 4 \sin \theta and y = 3 \cos \theta, form an equation relating x and y that does not involve \sin \theta or \cos \theta.
Prove the following identities:
\dfrac{\cos x \tan x}{\sin x} = 1
\dfrac{1 - \sin ^{2}\left(x\right)}{\cos x} = \cos x
\dfrac{\sin \theta}{1 - \cos \theta} = \dfrac{1 + \cos \theta}{\sin \theta}
\left(\sin x + \cos x\right)^{2} = 1 + 2 \sin x \cos x
\sin A \cos A \tan A = \sin ^{2}\left(A\right)
\cos ^{4}\left(x\right) - \sin ^{4}\left(x\right) = 2 \cos ^{2}\left(x\right) - 1
5 \cos ^{2}\left(\theta\right) - 3 = 2 - 5 \sin ^{2}\left(\theta\right)
\dfrac{\left(1 + \sin \theta\right)^{2} + \cos ^{2}\left(\theta\right)}{1 + \sin \theta} = 2
\dfrac{\sin ^{2}\left(x\right) + \sin x \cos x}{\cos ^{2}\left(x\right) + \sin x \cos x} = \tan x
Prove the following identities:
\tan ^{2}\left(y\right) - \sin ^{2}\left(y\right) = \tan ^{2}\left(y\right) \sin ^{2}\left(y\right)
\sin ^{2}\left(a\right) - \sin ^{2}\left(b\right) + \cos ^{2}\left(a\right) \sin ^{2}\left(b\right) - \sin ^{2}\left(a\right) \cos ^{2}\left(b\right) = 0
\left(1 - \sin ^{2}\left(x\right)\right) \left(1 + \sin ^{2}\left(x\right)\right) = 2 \cos ^{2}\left(x\right) - \cos ^{4}\left(x\right)