Answer:
see explanation
Step-by-step explanation:
(3)
Given cosΘ = -
Then by Pythagoras' theorem the third side is 3 ( 3,4, 5 triangle )
Since Θ in second quadrant then sinΘ > 0
sinΘ =
Using the trigonometric identity
sin2Θ = 2sinΘcosΘ, then
sin2Θ = 2 × × -
= -
(4)
Using the trigonometric identity
cos(x - y) = cosxcosy + sinxsiny
note cos15° = cos(45 - 30)°
cos(45 - 30) = cos45cos30 + sin45sin30
= ( ×
) + (
×
)
=
Answer:
see explanation
Step-by-step explanation:
(4)
consider the left side
factor the numerator
cosx - cos³x = cosx(1 - cos²x)
cancel sinx on numerator/denominator
= cosxsinx =right side ⇒ verified
(5)
Consider the left side
expand the factors
(1 + cotΘ)² + (1 - cotΘ)²
= 1 + 2cotΘ + cot²Θ + 1 - 2cotΘ + cot²Θ
= 2 + 2cot²Θ
= 2(1 + cot²Θ) ← 1 + cot²Θ = cosec²Θ
= 2cosec²Θ = right side ⇒ verified
(6)
Consider the left side
the denominator simplifies to
cosxtanx = cosx × = sinx
= sinx( +
)
= +
= tanx + 1 = right side ⇒ verified