Show the Steps for (sec 𝜃 + tan 𝜃)(sec 𝜃 − tan 𝜃) = 1.

Question

Use trigonometric identities to transform the left side of the equation into the right side (0 < 𝜃 < 𝜋/2).

(sec 𝜃 + tan 𝜃)(sec 𝜃 − tan 𝜃) = 1

Answer 1

We must simplify the left side of the identity to match the right side.

So

(sec 𝜃 + tan 𝜃)(sec 𝜃 − tan 𝜃) = 1

multiplying the binomials

sec² 𝜃 − tan² 𝜃 = 1

Converting to Sines and Cosines

1 - sin ² 𝜃 = 1

cos² 𝜃 cos ² 𝜃

Putting them over the common denominator

1 - sin ² 𝜃 = 1

cos ² 𝜃

Using the identity sin² 𝜃 + cos ² 𝜃 = 1, then cos ² 𝜃 = 1 - sin² 𝜃

Substituting in the numerator and reducing

cos ² 𝜃 = 1

cos ² 𝜃

1 = 1

Answer 2

(sec 𝜃 + tan 𝜃)(sec 𝜃 − tan 𝜃) = 1

Verify by converting the LHS into the RHS

Step 1: Since (a + b)(a – b) = a² – b² this means that

(sec 𝜃 + tan 𝜃)(sec 𝜃 − tan 𝜃) = sec² 𝜃 − tan² 𝜃

Step 2: sec² 𝜃 = 1 + tan² 𝜃 is a know fundamental trig identity

This means we can substitute 1 + tan² 𝜃 for sec² 𝜃 in the last equation giving

sec² 𝜃 − tan² 𝜃 = 1 + tan² 𝜃 − tan² 𝜃

Step 3: Simplify by canceling the tan² 𝜃 terms, proving that

(sec 𝜃 + tan 𝜃)(sec 𝜃 − tan 𝜃) = sec² 𝜃 − tan² 𝜃 = 1 + tan² 𝜃 − tan² 𝜃 = 1

thus verifying that the given equation is true.

Answer 3

Answer 4

Hello,

Please let me know if this works well for you, or if you'd like more insight on this problem.

Using your Pythagorean identities,

We know the following:

sin²(x) + cos²(x) = 1.

Lets go ahead and multiply both sides by (1/cos²(x)).

So, now we have:

--> [1/cos²(x)][sin²(x) + cos²(x)] = 1(1/cos²(x)) Equation (1)

----> sin²x / cos²(x) + cos²(x)/cos²(x) = 1/cos²(x) Equation (2)

Now, recall the identities:

tan(x) = sin(x)/cos(x) sec(x) = 1/cos(x)

After we square both sides of the equations, we now have the identities:

tan²(x) = sin²(x)/cos²(x) sec²(x) = 1/cos²(x)

Using these identities, we can further simplify equation (2) :

----> sin²x / cos²(x) + cos²(x)/cos²(x) = 1/cos²(x) Equation (2)

------> tan²(x) + 1 = sec²(x) Equation (3)

Subtract tan²(x) on both sides of the equation:

--------> 1 = sec²(x) - tan²(x) Equation (4)

Using algebra, we can foil out Equation (4):

----------> 1 = ( sec(x) + tan(x) ) ( sec(x) - tan(x) ) Equation (5)

Please let me know if this is well.

Best wishes.