1. sec α - cos α = sin α tan α
Start with the left side:
sec α - cos α = (1/cos α) - cos α
= (1 - cos²α) / cos α
= sin²α / cos α
= sin α * (sin α / cos α)
= sin α tan α
2. cos x / (1 - sin x) = (1 + sin x) / cos x
Start with the left side:
cos x / (1 - sin x)
Multiply numerator and denominator by (1 + sin x):
[cos x (1 + sin x)] / [(1 - sin x)(1 + sin x)]
= [cos x (1 + sin x)] / (1 - sin²x)
= [cos x (1 + sin x)] / cos²x
= (1 + sin x) / cos x
3. (sec x - tan x)(csc x + 1) = cot x
Start with the left side:
(sec x - tan x)(csc x + 1)
= [(1/cos x) - (sin x/cos x)] * [(1/sin x) + 1]
= [(1 - sin x)/cos x] * [(1 + sin x)/sin x]
= [(1 - sin²x)/(cos x sin x)]
= [cos²x/(cos x sin x)]
= cos x / sin x
= cot x
4. [1 - (cos θ - sin θ)²] / cos θ = 2 sin θ
Start with the left side:
Expand the square: (cos θ - sin θ)² = cos²θ - 2 sin θ cos θ + sin²θ
So, 1 - (cos²θ - 2 sin θ cos θ + sin²θ) = 1 - (1 - 2 sin θ cos θ) = 2 sin θ cos θ
Now divide by cos θ: (2 sin θ cos θ) / cos θ = 2 sin θ
5. 1 + cos θ = sin²θ / (1 - cos θ)
Start with the right side:
sin²θ / (1 - cos θ) = (1 - cos²θ) / (1 - cos θ)
= [(1 - cos θ)(1 + cos θ)] / (1 - cos θ)
= 1 + cos θ
6. tan²x - sin²x = tan²x sin²x
Start with the left side:
tan²x - sin²x = (sin²x / cos²x) - sin²x
= sin²x (1/cos²x - 1)
= sin²x ((1 - cos²x) / cos²x)
= sin²x (sin²x / cos²x)
= tan²x sin²x
7. 2 cos(2θ) + 4 sin²θ = 2
Start with the left side:
Use identity: cos(2θ) = 1 - 2 sin²θ
So, 2(1 - 2 sin²θ) + 4 sin²θ = 2 - 4 sin²θ + 4 sin²θ = 2
8. csc x / (cot x + tan x) = cos x
Start with the left side:
csc x / (cot x + tan x) = (1/sin x) / [(cos x/sin x) + (sin x/cos x)]
= (1/sin x) / [(cos²x + sin²x) / (sin x cos x)]
= (1/sin x) / [1 / (sin x cos x)]
= (1/sin x) * (sin x cos x)
= cos x
Parent Tip: Review the logic above to help your child master the concept of simplifying trig identities worksheet.
