This is a topic under Grade 11 Pre-Calculus. This will discuss proving and simplifying Trigonometric Identities.

1. Trigonometric
Identities
Simplifying and Proving

2. Classification of Equations
• Conditional Equation – only true for some values of the variable.
Ex. cos 𝜃 = sin 𝜃 is true for 𝜃 = 45° but not for 𝜃 = 30°
• Identity – true for all values of the variable.
Ex. tan 𝜃 cos 𝜃 = sin 𝜃

3. Trigonometric Identities

4. EXAMPLES: PROVE THE FOLLOWING
2.
𝑡𝑎𝑛𝜃
𝑠𝑖𝑛𝜃
= 𝑠𝑒𝑐𝜃
3. 𝑐𝑠𝑐𝜃sin𝜃 = cos𝜃𝑠𝑒𝑐𝜃
4.
𝑠𝑖𝑛𝜃
𝑡𝑎𝑛𝜃
1
𝑠𝑒𝑐𝜃
= 1 − 𝑠𝑖𝑛2
𝜃
5. 𝑡𝑎𝑛2
𝑥 = 𝑠𝑖𝑛2
𝑥𝑐𝑜𝑠−2
𝑥
6.
1−𝑐𝑜𝑠2 𝜃
𝑠𝑖𝑛2 𝜃
= 𝑐𝑜𝑠2
𝜃 + 𝑠𝑖𝑛2
𝜃
7. 𝑐𝑠𝑐2
𝜃 + 𝑠𝑒𝑐2
𝜃 = (1 − 𝑐𝑜𝑠2
𝜃)
−1
(1 − 𝑠𝑖𝑛2
𝜃)
−1

5. PROVE THE FOLLOWING EQUATIONS

6. 1. 𝑐𝑜𝑠2
𝑥 − 𝑠𝑖𝑛2
𝑥 = 𝑐𝑜𝑠4
𝑥 − 𝑠𝑖𝑛4
𝑥
2. 𝑡𝑎𝑛𝑥 +
1
𝑡𝑎𝑛𝑥
=
1
𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥
3.
𝑠𝑖𝑛2 𝑥
1−𝑐𝑜𝑠𝑥
= 1 + 𝑐𝑜𝑠𝑥
4.
tan 𝜃 sin 𝜃
cot 𝜃 cos 𝜃
= 𝑡𝑎𝑛3
𝜃

7. SIMPLIFY THE FOLLOWING STATEMENTS

8. 1.
sec 𝑥−cos 𝑥
sec 𝑥
2. (tan 𝑥 + cot 𝑥) sin x cos x
3.
1+csc 𝑥
cos 𝑥+cot 𝑥
4.
𝑐𝑜𝑠2 𝑥
1−sin 𝑥
5.
(1−𝑐𝑜𝑠2 𝑥)2
𝑠𝑖𝑛2 𝑥+𝑠𝑖𝑛2 𝑥𝑐𝑜𝑠𝑥