Is cos^2(x)tan^2(x)csc(x)=sin(x) ?

The answer to whether cos^2(x)tan^2(x)csc(x)=sin(x) is True

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prove cos^2(x)tan^2(x)csc(x)=sin(x)

prove

cos^{2} (x) tan^{2} (x) csc (x) = sin (x)

True

Solution steps

cos^{2} (x) tan^{2} (x) csc (x) = sin (x)

Manipulating left side

cos^{2} (x) tan^{2} (x) csc (x)

Express with sin, cos

cos^{2} (x) csc (x) tan^{2} (x)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

= \frac{1 \cdot cos ^{2} ( x ) ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{sin ( x )}

Multiply:

1 \cdot cos^{2} (x) = cos^{2} (x)

= \frac{( \frac{s i n ( x )}{c o s ( x )} ) ^{2} cos ^{2} ( x )}{sin ( x )}

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

(\frac{sin ( x )}{cos ( x )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

= \frac{\frac{s i n ^{2} ( x )}{c o s ^{2} ( x )} cos ^{2} ( x )}{sin ( x )}

Multiply

cos^{2} (x) \frac{sin ^{2} ( x )}{cos ^{2} ( x )} : sin^{2} (x)

cos^{2} (x) \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

Multiply fractions:

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

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

Cancel the common factor:

cos^{2} (x)

= sin^{2} (x)

= \frac{sin ^{2} ( x )}{sin ( x )}

Cancel the common factor:

sin (x)

= sin (x)

= sin (x)

= sin (x)

We showed that the two sides could take the same form

\Rightarrow True

7 cos^{2} (B) - 7 sin^{2} (B) = 14 cos^{2} (B) - 7

sec^{2} (x) (sec (x) - cos (x)) = tan^{2} (x)

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

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

2 cos (3 x) = cot (3 x)

Is cos^2(x)tan^2(x)csc(x)=sin(x) ?
The answer to whether cos^2(x)tan^2(x)csc(x)=sin(x) is True