How do you simplify $\frac{{csc}^{2} (θ)}{1 + {tan}^{2} (θ)}$ $csc^2(theta) / (1 + tan^2(theta) )$ ?

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How do you simplify $\frac{{csc}^{2} (θ)}{1 + {tan}^{2} (θ)}$ $csc^2(theta) / (1 + tan^2(theta) )$ ?

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Answer 1 · 2018-06-16T15:22:39

${cot}^{2} θ$ $cot^2theta$

Explanation:

$using the trigonometric identities$ $"using the "color(blue)"trigonometric identities"$

$∙ x 1 + {tan}^{2} θ = {sec}^{2} θ$ $•color(white)(x)1+tan^2theta=sec^2theta$

$∙ x csc θ = \frac{1}{sin θ} and sec θ = \frac{1}{cos θ}$ $•color(white)(x)csctheta=1/sintheta" and "sectheta=1/costheta$

$= \frac{{csc}^{2} θ}{{sec}^{2} θ}$ $=csc^2theta/sec^2theta$

$= \frac{\frac{1}{{sin}^{2} θ}}{\frac{1}{{cos}^{2} θ}}$ $=(1/sin^2theta)/(1/cos^2theta)$

$= \frac{1}{{sin}^{2} θ} \times {cos}^{2} θ = \frac{{cos}^{2} θ}{{sin}^{2} θ} = {cot}^{2} θ$ $=1/sin^2thetaxxcos^2theta=cos^2theta/sin^2theta=cot^2theta$

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How do you simplify $\frac{{csc}^{2} (θ)}{1 + {tan}^{2} (θ)}$ $csc^2(theta) / (1 + tan^2(theta) )$ ?

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