If - - - - -2- then show that cos 2- cos-sin- cos-sin- sin 2-cos2- cos-sin- cos-sin- sin 2- - 0

θ ϕ = π2 = cos^2 θ amp; cos θ .sin θ cos θ . sin θ amp; sin^2 θ cos^2 ϕ amp; cos ϕ . sin ϕ cos ϕ . sin ϕ amp; sin^2 ϕ = co ...

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Definition of a Determinant

If θ - ϕ = ...

Question

If $θ - ϕ = \frac{π}{2}$ , then show that $∣ ∣ ∣ \begin{matrix} {cos}^{2} θ & cos θ sin θ cos θ sin θ & {sin}^{2} θ \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} {cos}^{2} ϕ & cos ϕ sin ϕ cos ϕ sin ϕ & {sin}^{2} ϕ \end{matrix} ∣ ∣ ∣ = 0$

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θ - ϕ = π / 2

[\begin{matrix} {cos}^{2} θ & cos θ sin θ cos θ sin θ & {sin}^{2} θ \end{matrix}] [\begin{matrix} {cos}^{2} φ & cos φ sin φ cos φ sin φ & {sin}^{2} φ \end{matrix}] = 0

3 {sin}^{2} θ + 2 {sin}^{2} ϕ = 1

3 sin 2 θ = 2 sin 2 ϕ

0 < θ < \frac{π}{2}

0 < ϕ < \frac{π}{2}

θ + 2 ϕ

cot θ = \frac{1}{2}

sec ϕ = - \frac{5}{3}

θ \in (π, \frac{3 π}{2})

ϕ \in (\frac{π}{2}, π)

cot (θ - ϕ)

sin θ = \frac{1}{2}, cos ϕ = 1

0 < θ < \frac{π}{2}

0 < ϕ \leq \frac{π}{2}

(cot (θ + 2 ϕ))^{2}

0 < ϕ < \frac{π}{2}

x = \sum_{n = 0}^{\infty} {cos}^{2 n} ϕ, y = \sum_{n = 0}^{\infty} {sin}^{2 n} ϕ

z = \sum_{n = 0}^{\infty} {cos}^{2 n} ϕ {sin}^{2 n} ϕ

x y z = x y + z

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