Is (cos^2(x)+1)(sec^2(x))=tan^2(x)+2 ?

The answer to whether (cos^2(x)+1)(sec^2(x))=tan^2(x)+2 is True

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prove (cos^2(x)+1)(sec^2(x))=tan^2(x)+2

prove

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

True

Solution steps

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

Manipulating right side

tan^{2} (x) + 2

Rewrite using trig identities

tan^{2} (x) + 2

Use the Pythagorean identity:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= sec^{2} (x) - 1 + 2

Simplify

= sec^{2} (x) + 1

= sec^{2} (x) + 1

Express with sin, cos

1 + sec^{2} (x)

Use the basic trigonometric identity:

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

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

Simplify

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

1 + (\frac{1}{cos ( 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 )}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

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

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

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

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

Use the basic trigonometric identity:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

3 - 3 (cos (x) - sin (x))^{2} = - 3 sin (2 x)

\frac{cot ( T ) - tan ( T )}{sin ( T ) + cos ( T )} = csc (T) - sec (T)

csc (\frac{π}{2} - x) cos (x) = 1

1 - 2 cos^{2} (4) = 2 sin^{2} (4) - 1

\frac{tan ( θ )}{cot ( θ ) + tan ( θ )} = sin^{2} (θ)

Is (cos^2(x)+1)(sec^2(x))=tan^2(x)+2 ?
The answer to whether (cos^2(x)+1)(sec^2(x))=tan^2(x)+2 is True