Is (1-cos(2a))/(1+cos(2a))=tan^2(a) ?

The answer to whether (1-cos(2a))/(1+cos(2a))=tan^2(a) is True

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prove (1-cos(2a))/(1+cos(2a))=tan^2(a)

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

\frac{1 - cos ( 2 a )}{1 + cos ( 2 a )}

Rewrite using trig identities

\frac{1 - cos ( 2 a )}{1 + cos ( 2 a )}

Use the Double Angle identity:

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

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

Simplify

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

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

Add the numbers:

1 + 1 = 2

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

Expand

1 - (1 - 2 sin^{2} (a)) : 2 sin^{2} (a)

1 - (1 - 2 sin^{2} (a))

- (1 - 2 sin^{2} (a)) : - 1 + 2 sin^{2} (a)

- (1 - 2 sin^{2} (a))

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

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

1 - 1 = 0

= 2 sin^{2} (a)

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

Factor

- 2 sin^{2} (a) + 2 : 2 (- sin^{2} (a) + 1)

- 2 sin^{2} (a) + 2

Rewrite as

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

Factor out common term

2

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

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

Divide the numbers:

\frac{2}{2} = 1

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

Remove parentheses:

(- a) = - a

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

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

Use the Pythagorean identity:

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

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

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

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

Use the basic trigonometric identity:

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

tan (a) tan (a)

Simplify

tan (a) tan (a) : tan^{2} (a)

tan (a) tan (a)

Apply exponent rule:

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

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

= tan^{1 + 1} (a)

Add the numbers:

1 + 1 = 2

= tan^{2} (a)

tan^{2} (a)

tan^{2} (a)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is (1-cos(2a))/(1+cos(2a))=tan^2(a) ?
The answer to whether (1-cos(2a))/(1+cos(2a))=tan^2(a) is True