Is ((1+cos(2y)))/(sin^2(y))=2cot^2(y) ?

The answer to whether ((1+cos(2y)))/(sin^2(y))=2cot^2(y) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove ((1+cos(2y)))/(sin^2(y))=2cot^2(y)

prove

\frac{( 1 + cos ( 2 y ) )}{sin ^{2} ( y )} = 2 cot^{2} (y)

True

Solution steps

\frac{1 + cos ( 2 y )}{sin ^{2} ( y )} = 2 cot^{2} (y)

Manipulating left side

\frac{1 + cos ( 2 y )}{sin ^{2} ( y )}

Rewrite using trig identities

\frac{1 + cos ( 2 y )}{sin ^{2} ( y )}

Use the Double Angle identity:

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

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

1 + 2 cos^{2} (y) - 1 = 2 cos^{2} (y)

1 + 2 cos^{2} (y) - 1

Group like terms

= 2 cos^{2} (y) + 1 - 1

1 - 1 = 0

= 2 cos^{2} (y)

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

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

Rewrite using trig identities

= \frac{2 cos ( y )}{sin ( y )} \cdot \frac{cos ( y )}{sin ( y )}

Use the basic trigonometric identity:

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

\frac{2 cos ( y ) cot ( y )}{sin ( y )}

= 2 cot (y) \frac{cos ( y )}{sin ( y )}

Use the basic trigonometric identity:

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

2 cot (y) cot (y)

Simplify

2 cot (y) cot (y) : 2 cot^{2} (y)

2 cot (y) cot (y)

Apply exponent rule:

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

cot (y) cot (y) = cot^{1 + 1} (y)

= 2 cot^{1 + 1} (y)

Add the numbers:

1 + 1 = 2

= 2 cot^{2} (y)

2 cot^{2} (y)

2 cot^{2} (y)

We showed that the two sides could take the same form

\Rightarrow True

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

4 cos^{2} (B) - 4 sin^{2} (B) = 4 - 8 sin^{2} (B)

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

sec^{2} (x) + 1 = \frac{1 + cos ^{2} ( x )}{cos ^{2} ( x )}

cos (4 A) = 8 cos^{4} (A) - 8 cos^{2} (A) + 1

Is ((1+cos(2y)))/(sin^2(y))=2cot^2(y) ?
The answer to whether ((1+cos(2y)))/(sin^2(y))=2cot^2(y) is True