Question

without using a calculator, find the value of \\(\sin^{2}(79^{\circ})+\cos^{2}(79^{\circ})-\frac{\sin(25^{\circ})}{\cos(65^{\circ})}\\).

Explanation:

Step1: Use the Pythagorean identity

By the Pythagorean identity $\sin^{2}\theta+\cos^{2}\theta = 1$, for $\theta = 79^{\circ}$, we have $\sin^{2}(79^{\circ})+\cos^{2}(79^{\circ})=1$.

Step2: Use the co - function identity

We know that $\cos(65^{\circ})=\sin(90^{\circ}-65^{\circ})=\sin(25^{\circ})$ (since $\cos\alpha=\sin(90^{\circ}-\alpha)$). So, $\frac{\sin(25^{\circ})}{\cos(65^{\circ})}=\frac{\sin(25^{\circ})}{\sin(25^{\circ})}=1$.

Step3: Calculate the value of the expression

Substitute the values from Step 1 and Step 2 into the original expression:

$$
LATEXBLOCK0
$$

Answer:

$0$