Prove that tan theta * sin theta = (1 - cos^2 theta)/(sqrt(1 - sin^2 theta))​

This identity holds as long as for all integer .

For the proof, make use of the fact that:

(definition of tangents,) and

(Pythagorean identity,) which is equivalent to .

Step-by-step explanation:

Assume that for all integer . This requirement ensures that the on the left-hand side takes a finite value. Doing so also ensures that the denominator on the right-hand side is non-zero.

Make use of the fact that to rewrite the left-hand side:

.

Apply the Pythagorean identity and to rewrite this fraction:

.

Hence, .

answer: (c) m∠qpo + (2x + 16)° = 180°

step-by-step explanation:

the opposite angles of a quadrilateral are supplementary.

so, ∠o + ∠q = 180°   and   ∠p + ∠r = 180°

since ∠r = 2x + 16°, we can use substitution as follows:

∠p + ∠r = 180°

∠p + (2x + 16)° = 180°