Verifying Trig Identities

Question

Verify each identity. Show your Work.

13) 4 𝑐𝑜𝑠²𝜃 + 3𝑠𝑖𝑛²𝜃 = 3 + 𝑐𝑜𝑠²𝜃

14) tan 𝜃 + cot 𝜃 = sec 𝜃 csc𝜃

Answer 1

13) Once we collect all the trig terms on one side of the equation we get

3cos²(θ) + 3sin²(θ) = 3

Factor out the common term on the left side and then remember the Pythagorean Identity for trig functions:

cos²(θ) + sin²(θ) = 1

14) Remember the Quotient Identities for trig functions:

tan(θ) = sin(θ) / cos(θ) and cot(θ) = cos(θ) / sin(θ)

Now, replace both terms on the left of the given identity

sin(θ) / cos(θ) + cos(θ) / sin(θ) = sec(θ) csc(θ)

Add the fractions on the left side using common denominators and then you need to remember the Pythagorean Identity and the definitions of secant and cosecant to finish.

Answer 2

13 ) LHS

4cos²θ + 3 (1 - cos²θ)

4cos²θ + 3 - 3 cos²θ

3 + cos²θ = RHS

14) LHS

tanθ + cotθ

sinθ/cosθ +cosθ/sinθ

(sin²θ + cos²θ) /cosθ sinθ

1/cosθsinθ

secθ cosecθ = RHS

Answer 3

Verification of an identity requires that only one side of the equation be altered.

4 cos² θ + 3 sin² θ = 3 + cos² θ

4 cos² θ + 3(1 - cos² θ) = 3 + cos² θ

4 cos² θ + 3 - 3 cos² θ = 3 + cos² θ

cos² θ + 3 = 3 + cos² θ

3 + cos² θ = 3 + cos² θ

QED