Question

Verify trigonometric equation by substituting identifies to match the right hand side of the equation to the left hand side of the equation. Show all work. Problem:-tan^2x + sec^2x=1

Answer 1

This identities are known as Pythagorean identities because this can be represented and solved using the pythagoras theorem. Let's remind the theorem:

$A^2+B^2=C^2$

Now, we also should know that the trigonometric functions can be written using the sides of a triangle. For the tangent and secant, we know:

$\begin{gathered} \tan x=\frac{B}{A} \\ \sec x=\frac{C}{A} \end{gathered}$

In the next step I will substitute this identities to the given equation:

$\begin{gathered} -\tan ^2x+\sec ^2x=1 \\ -(\frac{B}{A})^2+(\frac{C}{A})^2=1 \\ -\frac{B^2}{A^2}+\frac{C^2}{A^2}=1 \\ \frac{C^2-B^2}{A^2}=1 \\ C^2-B^2=A^2 \\ C^2=A^2+B^2 \end{gathered}$

Which satisfy the Pythagoras Theorem.