Trigonometric Fundamental Identities

Question

Simplify the expression:

A. sin x - sin³ x

B. sec x cos² x + sec x sin² x

C. cos x (cot x + tan x)

D. sin x csc x + sin² x sec^{2 x}

Answer 1

A). sin x - sin³x

Take out common term sin(x)

sin(x)[1 - sin²x]

Pythagorean Identities: sin²θ + cos²θ = 1⇒cos²θ = 1 - sin²θ

sin(x)cos²x.

Answer 2

B). sec(x)cos²(x) + sec(x)sin²(x)

Take out common term sec(x)

sec(x)[cos²(x) + sin²(x)]

Pythagorean Identities: sin²θ + cos²θ = 1

sec(x)(1) = sec(x).

Therefore sec(x)cos²(x) + sec(x)sin²(x) = sec(x).

Answer 3

C). cos(x)[cot(x) + tan(x)]

Quotient Identities: tanθ = sinθ/cosθ and cotθ = cosθ/sinθ

cos(x)[cos(x)/sin(x) + sin(x)/cos(x)]

Rewrite the expression with common denominator.

cos(x)[cos²(x)+ sin²(x)]/sin(x)cos(x)

Pythagorean Identities: sin²θ + cos²θ = 1

cos(x)[1]/sin(x)cos(x) = 1/sin(x)

Reciprocal identities: 1/sin(x) = coses(x)

Therefore cos(x)[cot(x) + tan(x)] = coses(x).

Answer 4

D). sin(x)coses(x) + sin²(x)sec²(x)

Reciprocal identities: coses(x) = 1/sin(x) and cos(x) = 1/sec(x).

= sin(x)[1/sin(x)] + sin²(x)[1/cos²(x)]

Quotient Identities: tanθ = sinθ/cosθ

= 1 + tan²(x)

Pythagorean Identities:1+ tan²θ = sec²θ.

= sec²(x).

Therefore cos(x)[cot(x) + tan(x)] = sec²(x).