If \left(\right. cosθ + secθ \left.\right) = \frac{5}{2} then (cos2θ + sec2θ) = ?

Foundation

Mathematics Foundation

Introduction to Trigonometry

Question

If $(cosθ + secθ) = \frac{5}{2}$ then (cos²θ + sec²θ) = ?

$\frac{33}{4}$

$\frac{21}{4}$

$\frac{17}{4}$

$\frac{29}{4}$

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Given: $(\cos θ + \sec θ) = \frac{5}{2}$ . We need to find $(\cos^{2} θ + \sec^{2} θ)$ .

Square both sides of the given equation:

${(\cos θ + \sec θ)}^{2} = {(\frac{5}{2})}^{2}$

Expanding the left side:

$\cos^{2} θ + \sec^{2} θ + 2 \cdot \cos θ \cdot \sec θ = \frac{25}{4}$

Since $\cos θ \cdot \sec θ = 1$ , we substitute:

$\cos^{2} θ + \sec^{2} θ + 2 = \frac{25}{4}$

Therefore, $\cos^{2} θ + \sec^{2} θ = \frac{25}{4} - 2 = \frac{25}{4} - \frac{8}{4} = \frac{17}{4}$

Final Answer: $\frac{17}{4}$