Mathematics
Assertion (A): (1 − cosec2 θ)(1 − sec2 θ) = 1
Reason (R): 1 + tan2 θ = sec2 θ and 1 + cot2 θ = cosec2 θ
A is true, R is false
A is false, R is true
Both A and R are true
Both A and R are false
Trigonometric Identities
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Answer
Solving L.H.S,
(1 − cosec2 θ)(1 − sec2 θ)
= (− cot2 θ)(− tan2 θ)
= (tan2 θ)
= 1
L.H.S = R.H.S
Therefore, assertion (A) is true.
1 + tan2 θ = sec2 θ and 1 + cot2 θ = cosec2 θ
These are standard trigonometric identities derived from the unit circle and the Pythagorean theorem.
Therefore, reason (R) is true.
Both A and R are true.
Hence, option 3 is the correct option.
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