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Teacher:	Prof. JM	Section:	AS Mathematics Grade 11
Date:	Week	Grade	Grade 11

Trigonometry

Learning Objectives		Understand the fundamental trigonometric ratios (sine, cosine, tangent) and their applications in solving right-angled triangles. Learn to derive and prove trigonometric identities to simplify complex expressions and solve equations. Apply trigonometric concepts to solve real-world problems, including those involving periodic functions, angles, and waveforms.
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Student Materials: Name:.......................Grade.................... Trigonometry Proving Trigonometric Identities Worksheet Question 1: Prove the identity: \[ \frac{1 - \cos^2(x)}{\sin(x)} = \sin(x) \] Question 2: Prove the identity: \[ 1 + \tan^2(x) = \sec^2(x) \] Question 3: Prove the identity: \[ \cos(x) \cdot \sec(x) = 1 \] Question 4: Prove the identity: \[ \frac{1}{\sin(x)} - \frac{1}{\cos(x)} = \frac{\sin(x) - \cos(x)}{\sin(x)\cos(x)} \] Question 5: Prove the identity: \[ \tan(x) \cdot \cos(x) = \sin(x) \] Question 6: Prove the identity: \[ \sin^2(x) = 1 - \cos^2(x) \] Question 7: Prove the identity: \[ \frac{\cos(x)}{1 + \sin(x)} = \frac{1 - \sin(x)}{\cos(x)} \] Question 8: Prove the identity: \[ \sin(2x) = 2 \sin(x) \cos(x) \] Question 9: Prove the identity: \[ 1 - \cos^2(x) = \sin^2(x) \] Question 10: Prove the identity: \[ \frac{1 - \sin(x)}{1 + \sin(x)} = \frac{\cos^2(x)}{1 + \sin(x)} \] Answers: Answer 1: \( \sin^2(x) \) Answer 2: \( \sec^2(x) \) Answer 3: \( 1 \) Answer 4: \( \frac{\sin(x) - \cos(x)}{\sin(x) \cos(x)} \) Answer 5: \( \sin(x) \) Answer 6: \( 1 - \cos^2(x) \) Answer 7: \( \frac{1 - \sin(x)}{\cos(x)} \) Answer 8: \( 2 \sin(x) \cos(x) \) Answer 9: \( \sin^2(x) \) Answer 10: \( \cos^2(x) \)	Student Materials: Show detailed steps here
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Student Materials:

Question 1: Prove the identity: \[ \frac{1 - \cos^2(x)}{\sin(x)} = \sin(x) \]

Question 2: Prove the identity: \[ 1 + \tan^2(x) = \sec^2(x) \]

Question 3: Prove the identity: \[ \cos(x) \cdot \sec(x) = 1 \]

Question 4: Prove the identity: \[ \frac{1}{\sin(x)} - \frac{1}{\cos(x)} = \frac{\sin(x) - \cos(x)}{\sin(x)\cos(x)} \]

Question 5: Prove the identity: \[ \tan(x) \cdot \cos(x) = \sin(x) \]

Question 6: Prove the identity: \[ \sin^2(x) = 1 - \cos^2(x) \]

Question 7: Prove the identity: \[ \frac{\cos(x)}{1 + \sin(x)} = \frac{1 - \sin(x)}{\cos(x)} \]

Question 8: Prove the identity: \[ \sin(2x) = 2 \sin(x) \cos(x) \]

Question 9: Prove the identity: \[ 1 - \cos^2(x) = \sin^2(x) \]

Question 10: Prove the identity: \[ \frac{1 - \sin(x)}{1 + \sin(x)} = \frac{\cos^2(x)}{1 + \sin(x)} \]

Student Materials:

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