Is (4sin^2(t))/(tan^2(t))=4cos^2(t) ?

The answer to whether (4sin^2(t))/(tan^2(t))=4cos^2(t) is True

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prove (4sin^2(t))/(tan^2(t))=4cos^2(t)

prove

\frac{4 sin ^{2} ( t )}{tan ^{2} ( t )} = 4 cos^{2} (t)

True

Solution steps

\frac{4 sin ^{2} ( t )}{tan ^{2} ( t )} = 4 cos^{2} (t)

Manipulating left side

\frac{4 sin ^{2} ( t )}{tan ^{2} ( t )}

Express with sin, cos

\frac{4 sin ^{2} ( t )}{tan ^{2} ( t )}

Use the basic trigonometric identity:

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

= \frac{4 sin ^{2} ( t )}{( \frac{s i n ( t )}{c o s ( t )} ) ^{2}}

Simplify

\frac{4 sin ^{2} ( t )}{( \frac{s i n ( t )}{c o s ( t )} ) ^{2}} : 4 cos^{2} (t)

\frac{4 sin ^{2} ( t )}{( \frac{s i n ( t )}{c o s ( t )} ) ^{2}}

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

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

Apply exponent rule:

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

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

= \frac{4 sin ^{2} ( t )}{\frac{s i n ^{2} ( t )}{c o s ^{2} ( t )}}

Apply the fraction rule:

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

= \frac{4 sin ^{2} ( t ) cos ^{2} ( t )}{sin ^{2} ( t )}

Cancel the common factor:

sin^{2} (t)

= 4 cos^{2} (t)

= 4 cos^{2} (t)

= 4 cos^{2} (t)

We showed that the two sides could take the same form

\Rightarrow True

\frac{csc ^{2} ( t )}{csc ^{2} ( t ) - 2} = sec (2 t)

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

sin (x) \cdot tan (x) + cos (x) = sec (x)

tan (2 x) = tan (x + x)

sin^{3} (x) = \frac{3}{4} sin (x) - \frac{1}{4} sin (3 x)

Is (4sin^2(t))/(tan^2(t))=4cos^2(t) ?
The answer to whether (4sin^2(t))/(tan^2(t))=4cos^2(t) is True