Is sec(b)-(tan(b))(sin(b))=cos(b) ?

The answer to whether sec(b)-(tan(b))(sin(b))=cos(b) is True

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prove sec(b)-(tan(b))(sin(b))=cos(b)

prove

sec (b) - (tan (b)) (sin (b)) = cos (b)

True

Solution steps

sec (b) - tan (b) sin (b) = cos (b)

Manipulating left side

sec (b) - tan (b) sin (b)

Express with sin, cos

sec (b) - sin (b) tan (b)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= \frac{1}{cos ( b )} - sin (b) tan (b)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{1}{cos ( b )} - sin (b) \frac{sin ( b )}{cos ( b )}

Simplify

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

\frac{1}{cos ( b )} - sin (b) \frac{sin ( b )}{cos ( b )}

sin (b) \frac{sin ( b )}{cos ( b )} = \frac{sin ^{2} ( b )}{cos ( b )}

sin (b) \frac{sin ( b )}{cos ( b )}

Multiply fractions:

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

= \frac{sin ( b ) sin ( b )}{cos ( b )}

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

sin (b) sin (b)

Apply exponent rule:

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

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

= sin^{1 + 1} (b)

Add the numbers:

1 + 1 = 2

= sin^{2} (b)

= \frac{sin ^{2} ( b )}{cos ( b )}

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

Apply rule

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

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

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

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

Rewrite using trig identities

\frac{1 - sin ^{2} ( b )}{cos ( b )}

Use the Pythagorean identity:

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

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

= \frac{cos ^{2} ( b )}{cos ( b )}

Cancel the common factor:

cos (b)

= cos (b)

= cos (b)

We showed that the two sides could take the same form

\Rightarrow True

p ro v e sec (- x) \cdot sin (x) = tan (x)

p ro v e \frac{cot ^{2} ( x ) - 1}{csc ^{2} ( x )} = cos (2 x)

p ro v e csc^{2} (x) - csc^{2} (x) \cdot cos^{2} (x) = 1

p ro v e cos (θ) - sin (θ) sin (2 θ) = cos (θ) cos (2 θ)

p ro v e 1 + (cot (A))^{2} = \cdot (csc (A))^{2}

Is sec(b)-(tan(b))(sin(b))=cos(b) ?
The answer to whether sec(b)-(tan(b))(sin(b))=cos(b) is True