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Question
Prove the following.
secθ (1 – sinθ) (secθ + tanθ) = 1
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Solution
secθ (1 – sinθ)(secθ + tanθ) = 1
LHS = secθ (1 – sinθ)(secθ + tanθ)
LHS = (secθ – secθ sinθ)(secθ + tanθ)
`"LHS" = (secθ – 1/cosθ × sinθ)(secθ + tanθ)`
`"LHS" = (secθ – sinθ/cosθ)(secθ + tanθ)`
`"LHS" = (secθ – tan θ)(secθ + tanθ) ...[(a + b)(a - b) = a^2 - b^2]`
`"LHS" = sec^2θ – tan^2θ ...{(1 + tan^2θ = sec^2θ),(∴ sec^2θ − tan^2θ = 1):}`
LHS = 1
RHS = 1
LHS = RHS
Hence proved.
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