If ΞΈ is an acute angle and 5 πππ ππ ΞΈ = 7, then evaluate π ππΞΈ + πππ 2ΞΈ β 1.

# If ΞΈ is an acute angle and 5 πππ ππ ΞΈ = 7, then evaluate π ππΞΈ + πππ 2ΞΈ β 1.

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### Solution:

We have been given that 5 πππ ππ ΞΈ = 7, where ΞΈ is an acute angle and we have to find the value of π ππΞΈ + πππ 2ΞΈ β 1.

$\begin{array}{r}5\mathrm{cosec}\mathrm{\Xi Έ}=7\\ \beta \mathrm{cosec}\mathrm{\Xi Έ}=\frac{7}{5}\end{array}$

We know that πππ ππ ΞΈ = 1 / π ππΞΈ

Therefore, π ππ ΞΈ = 5/7

We also know that ${\mathrm{sin}}^{2}\mathrm{\Xi Έ}+{\mathrm{cos}}^{2}\mathrm{\Xi Έ}=1$

$\begin{array}{l}\beta {\mathrm{cos}}^{2}\mathrm{\Xi Έ}=11-{\mathrm{sin}}^{2}\mathrm{\Xi Έ}\\ \beta {\mathrm{cos}}^{2}\mathrm{\Xi Έ}=1-{\left(\frac{5}{7}\right)}^{2}\\ {\mathrm{cos}}^{2}\mathrm{\Xi Έ}=\frac{24}{49}\end{array}$

Substituting these values in the expression π ππΞΈ + πππ 2 ΞΈ, we get

$\begin{array}{ll}\mathrm{sin}& \mathrm{\Xi Έ}+{\mathrm{cos}}^{2}\mathrm{\Xi Έ}-1=\frac{5}{7}+\frac{24}{49}-1=\frac{10}{49}\end{array}$

Therefore the value of the given expression is $\frac{10}{49}$.

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