Question

Prove the Identity:


`\large{ (cos( - \theta))/(1 - sin \theta) - (cos( \theta + \pi))/(1 + sin \theta) = (2)/(cos \theta) }`
Show your work, thanks! ​

Answer 1

`\boxed{\large{\bold{\textbf{\textsf{{\color{blue}{Answer}}}}}}:)}`

• 
  `\sf{(cos( - \theta))/(1 - sin \theta) - (cos( \theta + \pi))/(1 + sin \theta) }`

`\\`

[
`\tt{cos(-\theta)=cos\theta }` ]

• 
  `\sf{(cos \theta)/(1 - sin \theta) - (cos( \theta + \pi))/(1 + sin \theta)}`

`\\`

[
`\tt{cos(\theta+\pi)=-cos\theta }` ]

• 
  `\sf{(cos \theta)/(1 - sin \theta) - (-cos\theta )/(1 + sin \theta)}`

• 
  `\sf{(cos \theta)/(1 - sin \theta) + (cos\theta )/(1 + sin \theta)}`

`\\`

• 
  `\sf{cos\theta((1)/(1-sin\theta)+(1)/(1+sin\theta) ) }`

• 
  `\sf{cos\theta((1+sin\theta+1-sin\theta)/((1)^2-(sin\theta)^2 ))}`

• 
  `\sf{cos\theta(\frac{1\cancel{+sin\theta}+1\cancel{-sin\theta}}{(1)^2-(sin\theta)^2 })}`

• 
  `\sf{cos\theta((2)/(1-sin^2\theta ))}`

`\\`

[
`\tt{1-sin^2\theta=cos^2\theta }` ]

• 
  `\sf{cos\theta((2)/(cos^2\theta ))}`

• 
  `\sf{cos\theta((2)/(cos\theta×cos\theta ))}`

• 
  `\sf{\cancel{cos\theta}×(\frac{2}{cos\theta×\cancel{cos\theta} })}`

• 
  `\sf{(2)/(cos\theta) }`

`\sf{ }`

`\sf{ }`

`\therefore{ (cos( - \theta))/(1 - sin \theta) - (cos( \theta + \pi))/(1 + sin \theta) = (2)/(cos \theta) }`(proved)

Answer 2

Use properties:

• cos(-θ) = cosθ
• cos(θ + π) = - cosθ
• sin²θ + cos²θ = 1

Solution:

• cosθ/(1 - sinθ) - (-cosθ)/(1 + sinθ) =
• cosθ([1/(a- sinθ) + 1/(1 + sinθ)] =
• cosθ[(1 + sinθ + 1 - sinθ)/(1 - sin²θ)] =
• cosθ(2/cos²θ) =
• 2/cosθ

Proved