Question

Find an exact value of each expression, without using a calculator${\cos }^{2}40+{\cos }^{2}50$

Short answer

The simplified value is$1$

Step-by-step solution

Step 1. Given information 

The given expression is${\cos }^{2}40+{\cos }^{2}50$

Step 2. Convert into the same trigonometrical ratio

The given expression is ${\cos }^{2}40+{\cos }^{2}50$

Here. $\cos (90-\theta )=\sin \theta$

Now$$\begin{gathered} ={\cos }^{2}40+{\cos }^{2}50 \\ ={\cos }^2(90-50)+{\cos }^{2}50 \\ ={\sin }^{2}50+{\cos }^{2}50 \\ =1 \end{gathered}$$Where${\sin }^2\theta +{\cos }^2\theta =1$

Step 3. Conclusion

Hence, the simplified value is$1$