Question

Start with a trigonometric expression, and apply substitutions and algebraic processes to create equivalent expressions. Justify each step. Verify your identity graphically and algebraically.Use the checklist!

Answer 1

Consider the expression below:

cos²A - sin²A

To find the equivalent expression, remember that:

cos²A = 1 - sin²A

Substitute cos²A = 1 - sin²A into cos²A - sin²A

1 - sin²A - sin²A

Which simplifies into:

1 - 2sin²A

Therefore, we can conclude that cos²A - sin²A and 1 - 2sin²A are equivalent expressions

Numerical Verification:

Let A = 30

cos²(30) - sin²(30) = (0.866)² - (0.5)² = 0.5 (to the nearest 1 decimal place)

Similarly,

1 - 2sin²(30) = 1 - 2(0.5)² = 1 - (0.5) = 0.5

We can conclude from the numerical calculation above that cos²A - sin²A and 1 - 2sin²A are equivalent

Graphical verification:

Graph of cos²A - sin²A is plotted below

Graph of 1 - 2sin²A is plotted below:

As seen from the two graphs plotted above, cos²A - sin²A and 1 - 2sin²A are equivalent