if sec theta +tan theta =m then value of sec^4 theta -tan⁴theta -2sec theta tan theta is
Answers
Given : secθ + tanθ = m
To Find : Value of sec⁴θ - tan⁴θ - 2secθtanθ
Solution:
sec⁴θ - tan⁴θ - 2secθtanθ
= (sec²θ)² - (tan²θ)² - 2secθtanθ
using a² - b² = ( a + b)(a - b)
a = sec²θ , b = tan²θ
= (sec²θ + tan²θ) (sec²θ - tan²θ) - 2secθtanθ
sec²θ - tan²θ = 1
=> (sec²θ + tan²θ) (1) - 2secθtanθ
= sec²θ + tan²θ - 2secθtanθ
= (secθ - tanθ)²
secθ + tanθ = m
multiplying both sides by (secθ - tanθ)
=> (secθ + tanθ)(secθ - tanθ) = m (secθ - tanθ)
=> (sec²θ - tan²θ) = m (secθ - tanθ)
=> 1 = m (secθ - tanθ)
=> (secθ - tanθ) = 1/m
sec⁴θ - tan⁴θ - 2secθtanθ = (secθ - tanθ)²
=> sec⁴θ - tan⁴θ - 2secθtanθ = (1/m)²
=> sec⁴θ - tan⁴θ - 2secθtanθ = 1/m²
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