Is tan(a) 1/(sin^2(a))=csc(a)sec(a) ?

The answer to whether tan(a) 1/(sin^2(a))=csc(a)sec(a) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove tan(a) 1/(sin^2(a))=csc(a)sec(a)

prove

tan (a) \frac{1}{sin ^{2} ( a )} = csc (a) sec (a)

True

Solution steps

tan (a) \frac{1}{sin ^{2} ( a )} = csc (a) sec (a)

Manipulating left side

tan (a) \frac{1}{sin ^{2} ( a )}

Express with sin, cos

\frac{1}{sin ^{2} ( a )} tan (a)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

Multiply:

1 \cdot sin (a) = sin (a)

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

Cancel the common factor:

sin (a)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

sin (x) = \frac{1}{csc ( x )}

\frac{1}{cos ( a ) \frac{1}{c s c ( a )}}

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{s e c ( a )} \cdot \frac{1}{c s c ( a )}}

Simplify

\frac{1}{\frac{1}{s e c ( a )} \cdot \frac{1}{c s c ( a )}}

Multiply

\frac{1}{sec ( a )} \cdot \frac{1}{csc ( a )} : \frac{1}{sec ( a ) csc ( a )}

\frac{1}{sec ( a )} \cdot \frac{1}{csc ( a )}

Multiply fractions:

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

= \frac{1 \cdot 1}{sec ( a ) csc ( a )}

Multiply the numbers:

1 \cdot 1 = 1

= \frac{1}{sec ( a ) csc ( a )}

= \frac{1}{\frac{1}{s e c ( a ) c s c ( a )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{sec ( a ) csc ( a )}{1}

Apply rule

\frac{a}{1} = a

= sec (a) csc (a)

sec (a) csc (a)

sec (a) csc (a)

= csc (a) sec (a)

We showed that the two sides could take the same form

\Rightarrow True

(1 + csc (x)) (1 + csc (- x)) = - cot^{4} (x)

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

(csc (α) + cot (α)) (sec (α) - 1) = tan (α)

1 - 2 sin^{2} (θ) = 2 cos^{2} (θ) - 1

sin (\frac{π}{4}) = cos (\frac{π}{4})

Is tan(a) 1/(sin^2(a))=csc(a)sec(a) ?
The answer to whether tan(a) 1/(sin^2(a))=csc(a)sec(a) is True