Prove the following trigonometry identities

Question

1. (1 − 𝑐𝑜𝑠2𝜃)𝑐𝑠𝑐2𝜃 = 1
2.𝑠𝑒𝑐𝜃√(1 − 𝑠𝑖𝑛2𝜃) = 1 [5]
3. 𝑐𝑜𝑠𝜃
(1−𝑡𝑎𝑛𝜃)
+
𝑠𝑖𝑛𝜃
1−𝑐0𝑡𝜃 = 𝑠𝑖𝑛𝜃 + 𝑐𝑜𝑠𝜃

Answer 1

To prove the first trigonometry identity, we start with the left side of the equation:

(1 − cos^2θ) sinθ cos^2θ

Using the Pythagorean identity sin^2θ + cos^2θ = 1, we can rewrite cos^2θ as:

1 − sin^2θ

Substituting this back into the previous equation, we get:

(1 − cos^2θ) sinθ (1 − sin^2θ)

Expanding this multiplication gives:

sinθ − sin^3θ

Now, using the identity sin^3θ = sinθ(1 − cos^2θ), which can be derived from sin^2θ = 1 − cos^2θ, we can rewrite the previous equation as:

sinθ − sinθ(1 − cos^2θ)

Simplifying further gives:

sinθ − sinθ + sinθcos^2θ

Combining like terms, we obtain:

sinθcos^2θ

Since sinθcos^2θ is equal to (1/2)sinθ(2cos^2θ), which is equal to (1/2)(2sinθcosθcosθ) by using the double angle identity, we can rewrite sinθcos^2θ as:

(1/2)sin2θcosθ

Using the identity sin2θ = 2sinθcosθ, we get:

(1/2)(2sinθcosθ)cosθ

Simplifying further:

(1/2)(2sinθcos^2θ)

Now, using the identity cos^2θ = 1 − sin^2θ, we can rewrite cos^2θ as:

(1/2)(2sinθ(1 − sin^2θ))

Expanding the multiplication gives:

sinθ − sin^3θ

This is equal to the right side of the equation, thus proving the identity:

(1 − cos^2θ)sinθcos^2θ = 1