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Answers
Step-by-step explanation:
(i) Given : tanθ + sinθ = m.
On squaring both sides, we get
⇒ (tanθ + sinθ)² = m²
⇒ tan²θ + sin²θ + 2tanθsinθ = m²
(ii) Given: tanθ - sinθ = n.
On squaring both sides, we get
⇒ tan²θ + sin²θ - 2tanθsinθ = n²
On subtracting (ii) from (i), we get
⇒ m² - n² = tan²θ + sin²θ + 2tanθsinθ - tan²θ - sin²θ + 2tanθsinθ
= 4tanθsinθ
= 4√tan²θsin²θ
= 4√(sin²θ/cos²θ)(1 - cos²θ)
= 4√sin²θ(1-cos²θ)/cos²θ
= 4√(sin²θ - sin²θcos²θ)/cos²θ
= 4√(sin²θ/cos²θ - sin²θ)
= 4√tan²θ - sin²θ
= 4√(tanθ + sinθ)(tanθ - sinθ)
= 4√mn.
Hope it helps!
tanθ-sinθ=n
m+n = tanθ+sinθ+tanθ-sinθ=2tanθ
m-n = tanθ+sinθ-tanθ+sinθ=2sinθ
mn = (tanθ+sinθ)(tanθ-sinθ)
= tan²θ-sin²θ
m²-n²
=(m+n)(m-n)
=2tanθ.2sinθ
=4sinθtanθ--------(1)
-----------4√mn-----------
=4√(tan²θ-sin²θ)
=4√(sin²θ/cos²θ-sin²θ)
=4√sin²θ(1/cos²θ-1)
=4sinθ√(1-cos²θ)/cos²θ
=4sinθ/cosθ√sin²θ [∵, sin²θ+cos²θ=1]
=4sinθtanθ-----------(1)
from--(1) and----(2)
m²-n² = 4√mn