2. cotФ = 
3. sec²Ф - tan²Ф = 1
4. 
= cos²∝
5. The value of sin x is 
6. cos 15° = 
7. tan(α + β) = 
8. sin(π - Ф) = sin Ф
9. cos 2Ф = 
10. sin 2Ф = 0.96
Step-by-step explanation:
2.
∵ cos Ф = 
∵ 90° < Ф < 180°
- That means Ф lies on the 2nd quadrant
∴ sin Ф is a positive value
∵ sin²Ф + cos²Ф = 1
∴ sin²Ф + ( )² = 1
∴ sin²Ф + 
= 1
- Subtract from both sides
∴ sin²Ф = 
- Take √ for both sides
∴ sinФ = 
∵ cotФ = cosФ ÷ sinФ
∴ cotФ = ÷
∴ cotФ =
3.
The expression is sec²Ф - tan²Ф
∵ tan²Ф = sec²Ф - 1
- Substitute tan²Ф by the right hand side in the expression
∴ sec²Ф - tan²Ф = sec²Ф - (sec²Ф - 1)
∴ sec²Ф - tan²Ф = sec²Ф - sec²Ф + 1
- Simplify the right hand side
∴ sec²Ф - tan²Ф = 1
4.
The expression is
∵ The numerator is sin²∝ + cos²∝
∵ sin²∝ + cos²∝ = 1
∴ The numerator = 1
∵ The denominator is tan²∝ + 1
∵ tan²∝ = 
∴ tan²∝ + 1 = + 1
- Change 1 to fraction 
∴ tan²∝ + 1 = +
- Add the two fractions
∴ tan²∝ + 1 = 
∵ sin²∝ + cos²∝ = 1
∴ tan²∝ + 1 = 
∴ The denominator =
∴ = 
- Remember denominator the denominator will be a numerator
∴ = cos²∝
5.
∵ tan x cos x =
∵ tan x = 
∴ × cos x =
- Simplify it by canceling cos x up with cos x down
∴ sin x =
∴ The value of sin x is
6.
∵ cos(Ф - ∝) = cosФ cos∝ + sinФ sin∝
∵ 45 - 30 = 15
∴ cos 15° = cos(45 - 30)°
- Use the rule above to find the exact value
∵ cos(45 - 30)° = cos 45° cos 30° + sin 45° sin 30°
∵ cos 45° = 
and sin 45° =
∵ cos 30° = 
and sin 30° =
∴ cos(45 - 30)° = × +
∴ cos(45 - 30)° = 
+ 
=
∴ cos 15° =
7.
∵ tan(α + β) = 
∵ cos α = 
and 0° < α < 90°
- That means α is in the 1st quadrant, then all trigonometry
ratios are positive
∵ sin²α + cos²α = 1
∴ sin²α + ( )² = 1
∴ sin²α + 
= 1
- Subtract from both sides
∴ sin²α = 
- Take √ for both sides
∴ sin α = 
∵ tan α = sin α ÷ cos α
∴ tan α = ÷
∴ tan α = 
∵ sin β = and 0° < β < 90°
∵ sin²β + cos²β = 1
∴ ( )² + cos²β = 1
∴ + cos²β = 1
- Subtract from both sides
∴ cos²β =
- Take √ for both sides
∴ cos β = 
∵ tan β = sin β ÷ cos β
∴ tan β = ÷
∴ tan β = 
- Substitute the values of tan α and tan β in the tan (α + β)
∵ tan(α + β) =
∴ tan(α + β) = 
∴ tan(α + β) =
8.
∵ sin(α - β) = sin α cos β - cos α sin β
∴ sin(π - Ф) = sin π cos Ф - cos π sin Ф
∵ sin π = 0 and cos π = -1
∴ sin(π - Ф) = (0) × cos Ф - (-1) × sin Ф
∴ sin(π - Ф) = 0 + sin Ф
∴ sin(π - Ф) = sin Ф
9.
∵ cos 2Ф = 2 cos²Ф - 1
∵ cos Ф =
∴ cos 2Ф = 2 ( )² - 1
∴ cos 2Ф = 2 × 
- 1
∴ cos 2Ф = - 1
∴ cos 2Ф =
10.
∵ sin 2Ф = 2 sinФ cosФ
∵ cosФ = 0.6 and 0° < Ф < 90°
- That means Ф is in the 1st quadrant and all its trigonometry
ratios are positive
∵ sin²Ф + cos²Ф = 1
∴ sin²Ф + (0.6)² = 1
∴ sin²Ф + 0.36 = 1
- Subtract 0.36 from both sides
∴ sin²Ф = 0.64
- Take √ for both sides
∴ sinФ = 0.8
- Substitute the values of sinФ and cosФ in the rule above
∴ sin 2Ф = 2(0.8)(0.6)
∴ sin 2Ф = 0.96
