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SIMPLIFYING TRIG EXPRESSIONS USING IDENTITIES

To simplify trigonometric identities, we have to remember the following problems.

Sometimes, we will be using the algebraic identities also to perform further simplification.

(a+b)2 = a2 + 2 ab + b2

(a-b)2 = a2 - 2 ab + b2

a2 - b2 = (a + b)(a - b)

Use Trigonometric Identities to write each expression in terms of a single trigonometric identity or a constant.

Problem 1 :

cot θ sin θ

Solution :

cot θ sin θ

= (cos θ/sin θ) ⋅ sin θ

= cos θ

Problem 2 :

1 - sin2𝜃sin2𝜃

Solution :

1 - sin2𝜃sin2𝜃= cos2𝜃sin2𝜃= cos 𝜃sin 𝜃2= cot2𝜃

Problem 3 :

sin θ sec θ

Solution :

sin θ sec θ

= sin θ × (1/cos θ)

= sin θ/cos θ

= tan θ

Problem 4 :

cos 𝜃 csc 𝜃 cot 𝜃

Solution :

cos 𝜃 csc 𝜃cot 𝜃= cos 𝜃 × 1sin 𝜃cot 𝜃= cos 𝜃 × 1sin 𝜃cos 𝜃sin 𝜃= cos 𝜃sin 𝜃cos 𝜃sin 𝜃= cos 𝜃sin 𝜃 × sin 𝜃cos 𝜃= 1

Problem 5 :

sec 𝜃 tan 𝜃

Solution :

sec 𝜃 tan 𝜃= 1cos 𝜃 tan 𝜃 =1cos 𝜃 sin 𝜃cos 𝜃= 1cos 𝜃 × cos 𝜃sin 𝜃= 1sin 𝜃= csc 𝜃

Problem 6 :

1 - sin2𝜃cot2𝜃

Solution :

1 - sin2𝜃cot2𝜃= cos2𝜃cot2𝜃= cos2𝜃cos2𝜃sin2𝜃= cos2𝜃 × sin2𝜃cos2𝜃= sin2𝜃

Problem 7 :

sin 𝜃 csc 𝜃 cot 𝜃

Solution :

sin 𝜃 csc 𝜃 cot 𝜃= sin 𝜃 × 1sin 𝜃 cot 𝜃= sin 𝜃 × 1sin 𝜃 cos 𝜃sin 𝜃= 1 cos 𝜃sin 𝜃= sin 𝜃cos 𝜃= tan 𝜃

Problem 8 :

cos θ/sec θ

Solution :

cos θ/sec θ

= cos θ/(1/cos θ)

= cos θ ⋅ cos θ

= cos2θ

Problem 9 :

tan 𝜃 + cot 𝜃 tan 𝜃

Solution :

tan 𝜃 + cot 𝜃 tan 𝜃= sin 𝜃cos 𝜃 + cos 𝜃sin 𝜃sin 𝜃cos 𝜃= sin2𝜃 + cos2𝜃sin 𝜃 cos 𝜃sin 𝜃cos 𝜃= 1sin 𝜃 cos 𝜃sin 𝜃cos 𝜃= cos 𝜃 sin 𝜃 cos 𝜃 sin 𝜃 = 1sin 2𝜃= csc2𝜃

Problem 10 :

cos2𝜃1 - sin 𝜃

Solution :

cos2𝜃1 - sin 𝜃= 1 - sin2𝜃 1 - sin 𝜃= (1 + sin 𝜃) (1 - sin 𝜃)1 - sin 𝜃= 1 + sin 𝜃

Problem 11 :

sec2𝜃 - 1sec2𝜃

Solution :

sec2𝜃 - 1sec2𝜃= tan2𝜃 sec2𝜃= sin2𝜃cos2𝜃1cos2𝜃= sin2𝜃cos2𝜃 × cos2𝜃= sin2𝜃

Problem 12 :

tan θ csc θ cos θ

Solution :

tan θ csc θ cos θ

= (sin θ/cos θ) ⋅ csc θ ⋅ cos θ

= (sin θ/cos θ) ⋅ (1/sin θ) ⋅ cos θ

= 1

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