Is (1-sin(v))/(cos(v))+(cos(v))/(1-sin(v))=2sec(v) ?

The answer to whether (1-sin(v))/(cos(v))+(cos(v))/(1-sin(v))=2sec(v) is True

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prove (1-sin(v))/(cos(v))+(cos(v))/(1-sin(v))=2sec(v)

prove

\frac{1 - sin ( v )}{cos ( v )} + \frac{cos ( v )}{1 - sin ( v )} = 2 sec (v)

True

Solution steps

\frac{1 - sin ( v )}{cos ( v )} + \frac{cos ( v )}{1 - sin ( v )} = 2 sec (v)

Manipulating left side

\frac{1 - sin ( v )}{cos ( v )} + \frac{cos ( v )}{1 - sin ( v )}

Simplify

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

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

Rewrite using trig identities

= \frac{2}{cos ( v )}

Rewrite using trig identities

2 sec (v)

We showed that the two sides could take the same form

\Rightarrow True

sec (θ) - tan (θ) = \frac{1}{sec ( θ ) + tan ( θ )}

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

9 sec^{2} (θ) - 5 tan^{2} (θ) = 5 + 4 sec^{2} (θ)

cot (2 θ) = cot (θ) - \frac{1}{2} sec (θ) csc (θ)

(csc (x) - cot (x)) (csc (x) + cot (x)) = 1

Is (1-sin(v))/(cos(v))+(cos(v))/(1-sin(v))=2sec(v) ?
The answer to whether (1-sin(v))/(cos(v))+(cos(v))/(1-sin(v))=2sec(v) is True