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Simplified expression of (sec θ + tan θ) (1 - sin θ) is
- a)sin2 θ
- b)cos2 θ
- c)tan2 θ
- d)cos θ
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
Simplified expression of (sec θ + tan θ) (1 - sin θ)...
Explanation:
Given expression: (sec θ + tan θ) (1 - sin θ)
Expand the expression:
(sec θ + tan θ) (1 - sin θ)
= sec θ - sec θ sin θ + tan θ - tan θ sin θ
= sec θ - sec θ sin θ + sin θ - sin2 θ
Using trigonometric identities:
sec θ = 1/cos θ
1 - sin2 θ = cos2 θ
Substitute the identities back into the expression:
sec θ - sec θ sin θ + sin θ - sin2 θ
= 1/cos θ - (1/cos θ)sin θ + sin θ - sin2 θ
= 1 - sin θ + sin θ - sin2 θ
= 1 - sin2 θ
= cos2 θ
Therefore, the simplified expression of (sec θ + tan θ) (1 - sin θ) is cos2 θ (option D).
Given expression: (sec θ + tan θ) (1 - sin θ)
Expand the expression:
(sec θ + tan θ) (1 - sin θ)
= sec θ - sec θ sin θ + tan θ - tan θ sin θ
= sec θ - sec θ sin θ + sin θ - sin2 θ
Using trigonometric identities:
sec θ = 1/cos θ
1 - sin2 θ = cos2 θ
Substitute the identities back into the expression:
sec θ - sec θ sin θ + sin θ - sin2 θ
= 1/cos θ - (1/cos θ)sin θ + sin θ - sin2 θ
= 1 - sin θ + sin θ - sin2 θ
= 1 - sin2 θ
= cos2 θ
Therefore, the simplified expression of (sec θ + tan θ) (1 - sin θ) is cos2 θ (option D).
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