Question
If tanθ + secθ = 5, find the value of sinθ.
Solution
Let t = tanθ, s = secθ. Given: t + s = 5 Use identity: sec²θ − tan²θ = 1 ⇒ (s − t)(s + t) = 1 So: (s − t) × 5 = 1 ⇒ s − t = 1/5 Add the two equations: s + t = 5 s − t = 1/5 2s = 5 + 1/5 = 25/5 + 1/5 = 26/5 s = 13/5 ⇒ secθ = 13/5 ⇒ cosθ = 5/13 Now, tanθ = t = s + t − s = 5 − 13/5 = (25 − 13)/5 = 12/5 So sinθ = tanθ × cosθ = (12/5) × (5/13) = 12/13.
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