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प्रश्न
Prove the following trigonometric identities.
`"cosec" theta sqrt(1 - cos^2 theta) = 1`
Prove the following:
`"cosec" theta sqrt(1 - cos^2 theta) = 1`
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उत्तर
We know that `sin^2 theta + cos^2 theta = 1`
So,
LHS = `"cosec" theta sqrt(1 - cos^2 theta)`
= `"cosec" theta sqrt (sin^2 theta)`
= cosec θ . sin θ
`1/sin theta xx sin theta`
= 1
= RHS hence proved.
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Solution:
In Δ ABC, ∠ABC = 90°, ∠C = θ°
AB2 + BC2 = `square` .....(Pythagoras theorem)
Divide both sides by AC2
`"AB"^2/"AC"^2 + "BC"^2/"AC"^2 = "AC"^2/"AC"^2`
∴ `("AB"^2/"AC"^2) + ("BC"^2/"AC"^2) = 1`
But `"AB"/"AC" = square and "BC"/"AC" = square`
∴ `sin^2 theta + cos^2 theta = square`
