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

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

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

prove

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

True

Solution steps

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

Manipulating left side

sec (a) - sin (a) tan (a)

Express with sin, cos

sec (a) - sin (a) tan (a)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

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

sin (a) sin (a)

Apply exponent rule:

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

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

= sin^{1 + 1} (a)

Add the numbers:

1 + 1 = 2

= sin^{2} (a)

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

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

Apply rule

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Cancel the common factor:

cos (a)

= cos (a)

= cos (a)

We showed that the two sides could take the same form

\Rightarrow True

\frac{cos ( t )}{sec ( t ) - tan ( t )} = 1 + sin (t)

sin (\frac{13 π}{2} + x) = cos (x)

tan^{2} (- θ) + 1 = sec^{2} (θ)

\frac{1 + csc ( β )}{cot ( β ) + cos ( β )} = sec (β)

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

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