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Question
Prove that: cot θ + tan θ = cosec θ·sec θ
Proof: L.H.S. = cot θ + tan θ
= `square/square + square/square` ......`[∵ cot θ = square/square, tan θ = square/square]`
= `(square + square)/(square xx square)` .....`[∵ square + square = 1]`
= `1/(square xx square)`
= `1/square xx 1/square`
= cosec θ·sec θ ......`[∵ "cosec" θ = 1/square, sec θ = 1/square]`
= R.H.S.
∴ L.H.S. = R.H.S.
∴ cot θ + tan θ = cosec·sec θ
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Solution
Proof: L.H.S. = cot θ + tan θ
= `bbcos θ/bbsin θ + bbsin θ/bbcos θ` ......`[∵ cot θ = bbcos θ/bbsin θ, tan θ = bb sinθ/bbcos θ]`
= `(bb(cos^2θ) + bb(sin^2θ))/(bbsin θ xx bbcos θ)` .....`[∵ bb(cos^2θ) + bb(sin^2θ) = 1]`
= `1/(bb sin θ xx bb cos θ)`
= `1/bb sin θ xx 1/bb cos θ`
= cosec θ·sec θ ......`[∵ "cosec" θ = 1/bb sin θ, sec θ = 1/bb cos θ]`
= R.H.S.
∴ L.H.S. = R.H.S.
∴ cot θ + tan θ = cosec·sec θ
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Proof: L.H.S. = sec θ + tan θ
= `1/square + square/square`
= `square/square` ......`(∵ sec θ = 1/square, tan θ = square/square)`
= `((1 + sin θ) square)/(cos θ square)` ......[Multiplying `square` with the numerator and denominator]
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