Is sin(t)(csc(t)-sin(t))=cos^2(t) ?

The answer to whether sin(t)(csc(t)-sin(t))=cos^2(t) is True

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prove sin(t)(csc(t)-sin(t))=cos^2(t)

prove

sin (t) (csc (t) - sin (t)) = cos^{2} (t)

True

Solution steps

sin (t) (csc (t) - sin (t)) = cos^{2} (t)

Manipulating left side

sin (t) (csc (t) - sin (t))

Express with sin, cos

(csc (t) - sin (t)) sin (t)

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

= (\frac{1}{sin ( t )} - sin (t)) sin (t)

Simplify

(\frac{1}{sin ( t )} - sin (t)) sin (t) : 1 - sin^{2} (t)

(\frac{1}{sin ( t )} - sin (t)) sin (t)

Join

\frac{1}{sin ( t )} - sin (t) : \frac{1 - sin ^{2} ( t )}{sin ( t )}

\frac{1}{sin ( t )} - sin (t)

Convert element to fraction:

sin (t) = \frac{sin ( t ) sin ( t )}{sin ( t )}

= \frac{1}{sin ( t )} - \frac{sin ( t ) sin ( t )}{sin ( t )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 - sin ( t ) sin ( t )}{sin ( t )}

1 - sin (t) sin (t) = 1 - sin^{2} (t)

1 - sin (t) sin (t)

sin (t) sin (t) = sin^{2} (t)

sin (t) sin (t)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (t) sin (t) = sin^{1 + 1} (t)

= sin^{1 + 1} (t)

Add the numbers:

1 + 1 = 2

= sin^{2} (t)

= 1 - sin^{2} (t)

= \frac{1 - sin ^{2} ( t )}{sin ( t )}

= \frac{- sin ^{2} ( t ) + 1}{sin ( t )} sin (t)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( 1 - sin ^{2} ( t ) ) sin ( t )}{sin ( t )}

Cancel the common factor:

sin (t)

= 1 - sin^{2} (t)

= 1 - sin^{2} (t)

= 1 - sin^{2} (t)

Rewrite using trig identities

1 - sin^{2} (t)

Use the Pythagorean identity:

1 = cos^{2} (x) + sin^{2} (x)

1 - sin^{2} (x) = cos^{2} (x)

= cos^{2} (t)

= cos^{2} (t)

We showed that the two sides could take the same form

\Rightarrow True

\frac{sin ( θ ) + sin ( 3 θ )}{cos ( θ ) + cos ( 3 θ )} = tan (2 θ)

1 + cos (\frac{3 π}{4}) = 1 - sin (\frac{3 π}{4})

cos (9 0^{\circ} - x) = sin (x)

sin (\frac{7 π}{6} + x) - cos (\frac{2 π}{3} + x) = 0

(1 + cos (a)) (1 - cos (a)) = sin^{2} (a)

Is sin(t)(csc(t)-sin(t))=cos^2(t) ?
The answer to whether sin(t)(csc(t)-sin(t))=cos^2(t) is True