Question

Please help, performance task: trigonometric identities

Answer 1

#a

• (1+cosx)(1-cosx)
• 1²-cos²x
• 1-cos²x
• sin²x

#2

• 1/cot²x-1/cos²x
• tan²x-sec²x
• -1

#c

• sec²(90-x)[sin²x-sin⁴x]
• csc²x[sin²x-sin⁴x]
• 1-sin²x
• cos²x

Answer 2

Trigonometric Identities used:

`\sin^2 \theta + \cos^2 \theta \equiv 1`

`\sec \theta \equiv (1)/(\cos \theta)`

`\cot \theta \equiv (1)/(\tan \theta)`

`\sec^2 \theta \equiv 1 + \tan^2 \theta`

`\sin(\theta)=\cos \left((\pi)/(2)-\theta\right)`

Part (a)

`(1+ \cos(x))(1-\cos(x))`

`=1-\cos(x)+\cos(x)-\cos^2(x)`

`=1-\cos^2(x)`

`= \sin^2(x)`

Part (b)

`(1)/(\cot^2(x))-(1)/(\cos^2(x))`

`=\tan^2(x)-\sec^2(x)`

`=\tan^2(x)-(1 + \tan^2(x))`

`=\tan^2(x)-1 - \tan^2(x)`

`=-1`

Part (c)

`\sec^2\left(( \pi )/(2)-x\right) \left[\sin^2(x)-\sin^4(x) \right]`

`=(1)/(\cos^2\left(( \pi )/(2)-x\right)) \left[\sin^2(x)-\sin^4(x) \right]`

`=(1)/(\sin^2(x)) \left[\sin^2(x)-\sin^4(x) \right]`

`= (\sin^2(x)-\sin^4(x))/(\sin^2(x))`

`= (\sin^2(x))/(\sin^2(x)) - (\sin^4(x))/(\sin^2(x))`

`= (\sin^2(x))/(\sin^2(x)) - ((\sin^2(x))(\sin^2(x)))/(\sin^2(x))`

`=1- \sin^2(x)`

`= \cos^2(x)`