Question

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Answer 1

Answer:

see explanation

Step-by-step explanation:

(4)

consider the left side

factor the numerator

cosx - cos³x = cosx(1 - cos²x)

$\frac{cosx(1-cos^2x)}{sinx}[/tex = [tex]\frac{cosxsin^2x}{sinx}$

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 × $\frac{sinx}{cosx}$ = sinx

$\frac{sin^2x(secx+cosecx)}{sinx}$

= sinx( $\frac{1}{cosx}$ + $\frac{1}{sinx}$)

= $\frac{sinx}{cosx}$ + $\frac{sinx}{sinx}$

= tanx + 1 = right side ⇒ verified