Is cos^2(α)sec^2(α)=1 ?

The answer to whether cos^2(α)sec^2(α)=1 is True

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prove cos^2(α)sec^2(α)=1

prove

cos^{2} (α) sec^{2} (α) = 1

True

Solution steps

cos^{2} (α) sec^{2} (α) = 1

Manipulating left side

cos^{2} (α) sec^{2} (α)

Express with sin, cos

cos^{2} (α) sec^{2} (α)

Use the basic trigonometric identity:

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

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

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

cos^{2} (α) (\frac{1}{cos ( α )})^{2}

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

(\frac{1}{cos ( α )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{cos ^{2} ( α )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( α )}

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

Multiply fractions:

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

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

Cancel the common factor:

cos^{2} (α)

= 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

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