SIMPLIFY COMPLEX FRACTIONS TRIGONOMETRY

Simplify the following :

Problem 1 :

(tan θ + cot θ) / tan θ

Solution :

Problem 2 :

cos²θ / (1 - sin θ)

Solution :

cos²θ / (1 - sin θ)

cos²θ = 1- sin²θ

1 can be written as 1². 1² - sin²θ is in the form of a² - b², the expansion of this will be (a + b) (a - b)

= 1- sin²θ / (1 - sin θ)

= (1 + sin θ) (1 - sin θ) / (1 - sin θ)

= (1 + sin θ)

Problem 3 :

(sec²θ - 1) / sec²θ

Solution :

(sec²θ - 1) / sec²θ ----(1)

Using Pythagorean identity,

sec²θ - tan²θ = 1

sec²θ - 1 =  tan²θ

Applying the value in (1), we get

= tan²θ  / sec²θ

= (sin²θ / cos²θ) / (1/cos²θ)

= (sin²θ / cos²θ) x (cos²θ/1)

= sin²θ

Problem 4 :

(tan θ + 1)²

Solution :

(a + b)² = a² + 2ab + b²

(tan θ + 1)² = (tan θ)² + 2(tan θ) (1) + 1²

= tan² θ + 2tan θ + 1

= 1 + tan² θ + 2 tan θ

= sec²θ + 2 tan θ

Problem 5 :

sin² θ - 2 sin θ + 1

Solution :

(a - b)² = a² - 2ab + b²

sin² θ - 2 sin θ + 1 = (sin θ)² - 2 sin θ x 1 + 1²

= (sin θ - 1)²

Problem 6 :

Solution :

Problem 7 :

tan θ - (sec² θ / tan θ)

Solution :

tan θ - (sec² θ / tan θ)

= tan θ - [(1/cos² θ) / (sin θ/cos θ)]

= tan θ - (1/cos² θ) x (cos θ/sin θ)

= tan θ - (1/sin θ cos θ)

= (sin θ /cos θ) - (1/sin θ cos θ)

= (sin² θ /sin θcos θ) - (1/sin θ cos θ)

= (sin² θ - 1) / sin θcos θ

= -cos² θ  / sin θ cos θ

= -cos θ  / sin θ 

= -cot θ

Problem 8 :

(tan θ / cot θ) + 1

Solution :

= (tan θ / cot θ) + 1

= [(sin θ  / cos θ) / (cos θ / sin θ)] + 1

= [(sin θ  / cos θ) x (sin θ / cos θ)] + 1

= 1 + 1

= 2

Problem 9 :

csc θ  sec θ - tan θ

Solution :

= csc θ  sec θ - tan θ

= (1/sin θ)(1/cos θ) - (sin θ / cos θ)

= (1/sin θ cos θ) - (sin θ / cos θ)

= (1/sin θ cos θ) - (sin θ / sin θ cos θ)

= (1/sin θ cos θ) - (sin² θ / sin θ cos θ)

= (1 - sin²θ) / sin θ cos θ

= cos²θ / sin θ cos θ

= cos θ / sin θ

= cot θ

Problem 10 :

sec²θ / (cot²θ + 1)

Solution :

= sec²θ / (cot²θ + 1)

= (1/cos²θ) / csc²θ

= (1/cos²θ) / (1/sin²θ)

= (1/cos²θ) x (sin²θ/1)

= (sin²θ/cos²θ)

= tan²θ