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7 – 2a: Verifying Trigonometric Identities

In this lesson, you will show that two expressions are equivalent. Your goal is to get the left hand side (LHS)

and the right hand side (RHS) to be equal.

You cannot:

Verify each of the identities below.

Type 1: Rewrite everything in terms of Sine and Cosine.

cot 𝑥 + 1 = csc 𝑥(cos 𝑥 + sin 𝑥)

Type 2: Split up the fraction.

tan 𝑡 − cot 𝑡

sin 𝑡 ∙ cos 𝑡= sec2 𝑡 − csc2 𝑡

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Type 3: Use a conjugate.

cos 𝑥

1 − sin 𝑥=

1 + sin 𝑥

cos 𝑥

Type 4: Combine fractions with a common denominator.

1

1 − sin 𝜃+

1

1 + sin 𝜃= 2 sec2 𝜃

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7 – 2b: Simplifying Trigonometric Identities

On October 4, 2006 Akira Haraguchi broke his own record by reciting the number pi to 100,000 decimal

places. To find out how long it took him, follow the directions below.

3 5 12 9 1 6 2 4 7 11 8 10

Simplify each trig expression. Find your answer in the list below. Write the letter associated with your

answer in the box that contains the question number. You may use answers more than once.

𝑬. 𝒔𝒆𝒄 𝒙 𝑵. 𝒄𝒔𝒄 𝒙 𝑺. 𝟏 𝑿. −𝟏 𝑹. 𝒄𝒐𝒔 𝒙

𝑯. 𝒕𝒂𝒏𝟐𝒙 𝑶. 𝒔𝒊𝒏𝟐𝒙 𝑻. 𝒄𝒐𝒕𝟐𝒙 𝑼. 𝒔𝒊𝒏 𝒙 𝑰. 𝒔𝒆𝒄𝟐𝒙

1. 𝑐𝑠𝑐 𝑥 𝑡𝑎𝑛 𝑥 2. 𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑡 𝑥 ∙ 𝑐𝑜𝑠 𝑥 3. 𝑐𝑠𝑐2𝑥 − 𝑐𝑜𝑡2𝑥

4. 𝑠𝑒𝑐2𝑥 − 𝑐𝑜𝑠2𝑥 ∙ 𝑠𝑒𝑐2𝑥 5. 𝑠𝑖𝑛2𝑥 + 𝑐𝑜𝑠2𝑥 + 𝑡𝑎𝑛2𝑥 6. 𝑐𝑜𝑠 𝑥 (1 + 𝑡𝑎𝑛2𝑥)

7. 𝑠𝑖𝑛 𝑥 ∙ 𝑐𝑠𝑐 𝑥 − 𝑐𝑜𝑠2𝑥 8. 𝑠𝑒𝑐 𝑥 − 𝑠𝑖𝑛 𝑥 ∙ 𝑡𝑎𝑛 𝑥 9. (𝑐𝑠𝑐 𝑥 + 1)(𝑐𝑠𝑐 𝑥 − 1)

10. 𝑠𝑖𝑛 𝑥

𝑐𝑜𝑠 𝑥 ∙ 𝑡𝑎𝑛𝑥 11. (𝑐𝑠𝑐 𝑥 + 𝑐𝑜𝑡 𝑥)(1 − 𝑐𝑜𝑠 𝑥) 12. (𝑡𝑎𝑛2𝑥 − 𝑠𝑒𝑐2𝑥)(𝑠𝑖𝑛2𝑥 + 𝑐𝑜𝑠2𝑥)

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7 – 2c: Verifying Identities

Verify each identity.

1. 𝑠𝑒𝑐2𝑥 − tan 𝑥 ∙ cot 𝑥 = 𝑡𝑎𝑛2𝑥

2. 7 𝑠𝑖𝑛 𝜃 +5 𝑐𝑜𝑠 𝜃

𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃= 7𝑠𝑒𝑐 𝜃 + 5 𝑐𝑠𝑐 𝜃

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3. 𝑠𝑖𝑛 𝑎

𝑐𝑠𝑐 𝑎+

𝑐𝑜𝑠 𝑎

𝑠𝑒𝑐 𝑎= 𝑐𝑠𝑐2𝑎 − 𝑐𝑜𝑡2𝑎

4. 1

𝑡𝑎𝑛𝑥 + 𝑠𝑒𝑐𝑥=

𝑐𝑜𝑠 𝑥

𝑠𝑖𝑛 𝑥 + 1

5. 1

csc 𝑥∙

cot 𝑥

cos 𝑥= sin2 𝑥 + cos2 𝑥