Trigonometric Identities
Establish each identity.
1. (a) \( \dfrac{\cot x}{\csc x}= \cos x \)
(b) \(\cos \theta \cdot \csc \theta \cdot \tan \theta = 1 \)
(c) \(\csc \theta \cdot \tan \theta = \sec \theta \)
SOLUTION
1. \( \sec\theta - \sec\theta \sin^2\theta = \cos\theta\)
SOLUTION
2. \( (\sin \theta + \cos \theta)^2 + (\sin \theta - \cos
\theta)^2 =2 \) SOLUTION
3. \( \displaystyle{ 1 -\frac{\sin^2\theta}{1+\cos\theta}=
\cos\theta }\)
SOLUTION
4. \( \displaystyle{ \tan\theta + \frac{\cos\theta}{1+\sin\theta}=\sec\theta} \)
SOLUTION
5. \( \displaystyle{ \frac{1-\cos\theta}{\sin
\theta}+\frac{\sin\theta}{1-\cos\theta} = 2\csc\theta} \)
SOLUTION
6. \( \displaystyle{ \frac{1}{1+\sin\theta}+\frac{1}{1-\sin\theta} =
2\sec^2\theta} \) SOLUTION
7. \( 5\cos^2\theta + 3\sin^2\theta = 3+2\cos^2\theta \)
SOLUTION
8. \( \displaystyle{ \csc u - \cot u = \frac{\sin u}{1+\cos u}}\)
SOLUTION
9. \( \displaystyle{ \frac{\sin\theta}{1-\cos\theta} =
\frac{1+\cos\theta}{\sin \theta} }\)
SOLUTION