Answer:
see explanation
Step-by-step explanation:
(4)
consider the left side
factor the numerator
cosx - cos³x = cosx(1 - cos²x)
[tex]\frac{cosx(1-cos^2x)}{sinx}[/tex = [tex]\frac{cosxsin^2x}{sinx}[/tex]
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 × [tex]\frac{sinx}{cosx}[/tex] = sinx
[tex]\frac{sin^2x(secx+cosecx)}{sinx}[/tex]
= sinx( [tex]\frac{1}{cosx}[/tex] + [tex]\frac{1}{sinx}[/tex])
= [tex]\frac{sinx}{cosx}[/tex] + [tex]\frac{sinx}{sinx}[/tex]
= tanx + 1 = right side ⇒ verified
