If θ is an acute angle and 5 𝑐𝑜𝑠𝑒𝑐 θ = 7, then evaluate 𝑠𝑖𝑛θ + 𝑐𝑜𝑠²θ − 1.

We have been given that 5 𝑐𝑜𝑠𝑒𝑐 θ = 7, where θ is an acute angle and we have to find the value of 𝑠𝑖𝑛θ + 𝑐𝑜𝑠²θ − 1.

$\begin{array}{r}5\mathrm{cosec}\theta =7\\ \Rightarrow \mathrm{cosec}\theta =\frac{7}{5}\end{array}$

We know that 𝑐𝑜𝑠𝑒𝑐 θ = 1 / 𝑠𝑖𝑛θ

Therefore, 𝑠𝑖𝑛 θ = 5/7

We also know that ${\sin }^2\theta +{\cos }^2\theta =1$

$\begin{array}{l}\Rightarrow {\cos }^2\theta =11-{\sin }^2\theta \\ \Rightarrow {\cos }^2\theta =1-{\left(\frac{5}{7}\right)}^2\\ {\cos }^2\theta =\frac{24}{49}\end{array}$

Substituting these values in the expression 𝑠𝑖𝑛θ + 𝑐𝑜𝑠² θ, we get

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

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