Is 2/(sec^2(x))+2/(csc^2(x))=2 ?

The answer to whether 2/(sec^2(x))+2/(csc^2(x))=2 is True

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prove 2/(sec^2(x))+2/(csc^2(x))=2

prove

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

True

Solution steps

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

Manipulating left side

\frac{2}{sec ^{2} ( x )} + \frac{2}{csc ^{2} ( x )}

Express with sin, cos

\frac{2}{csc ^{2} ( x )} + \frac{2}{sec ^{2} ( x )}

Use the basic trigonometric identity:

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

= \frac{2}{( \frac{1}{s i n ( x )} ) ^{2}} + \frac{2}{sec ^{2} ( x )}

Use the basic trigonometric identity:

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

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

Simplify

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

\frac{2}{( \frac{1}{s i n ( x )} ) ^{2}} + \frac{2}{( \frac{1}{c o s ( x )} ) ^{2}}

\frac{2}{( \frac{1}{s i n ( x )} ) ^{2}} = 2 sin^{2} (x)

\frac{2}{( \frac{1}{s i n ( x )} ) ^{2}}

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= \frac{2}{\frac{1}{s i n ^{2} ( x )}}

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= 2 sin^{2} (x)

\frac{2}{( \frac{1}{c o s ( x )} ) ^{2}} = 2 cos^{2} (x)

\frac{2}{( \frac{1}{c o s ( x )} ) ^{2}}

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= \frac{2}{\frac{1}{c o s ^{2} ( x )}}

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= 2 cos^{2} (x)

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

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

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

Factor

2 cos^{2} (x) + 2 sin^{2} (x) : 2 (cos^{2} (x) + sin^{2} (x))

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

Factor out common term

2

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

= (cos^{2} (x) + sin^{2} (x)) \cdot 2

Rewrite using trig identities

(cos^{2} (x) + sin^{2} (x)) \cdot 2

Use the Pythagorean identity:

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

= 1 \cdot 2

Simplify

= 2

= 2

We showed that the two sides could take the same form

\Rightarrow True

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

sin (2 θ) = 2 sin^{2} (θ) - 1

sec (x) - sin (x) = cot (x) cos (x)

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

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

Is 2/(sec^2(x))+2/(csc^2(x))=2 ?
The answer to whether 2/(sec^2(x))+2/(csc^2(x))=2 is True