Is 1+sec(θ)tan(θ)sin(θ)=sec^2(θ) ?

The answer to whether 1+sec(θ)tan(θ)sin(θ)=sec^2(θ) is True

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prove 1+sec(θ)tan(θ)sin(θ)=sec^2(θ)

prove

1 + sec (θ) tan (θ) sin (θ) = sec^{2} (θ)

True

Solution steps

1 + sec (θ) tan (θ) sin (θ) = sec^{2} (θ)

Manipulating left side

1 + sec (θ) tan (θ) sin (θ)

Rewrite using trig identities

1 + sec (θ) tan (θ) sin (θ)

Express with sin, cos

sec (θ) sin (θ)

Use the basic trigonometric identity:

sec (θ) = \frac{1}{cos ( θ )}

= \frac{1}{cos ( θ )} sin (θ)

Simplify

\frac{1}{cos ( θ )} sin (θ) : \frac{sin ( θ )}{cos ( θ )}

\frac{1}{cos ( θ )} sin (θ)

Multiply fractions:

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

= \frac{1 sin ( θ )}{cos ( θ )}

Multiply:

1 \cdot sin (θ) = sin (θ)

= \frac{sin ( θ )}{cos ( θ )}

= \frac{sin ( θ )}{cos ( θ )}

= 1 + \frac{sin ( θ )}{cos ( θ )} tan (θ)

Use the basic trigonometric identity:

\frac{sin ( θ )}{cos ( θ )} = tan (θ)

= 1 + tan (θ) tan (θ)

tan (θ) tan (θ) = tan^{2} (θ)

tan (θ) tan (θ)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

tan (θ) tan (θ) = tan^{1 + 1} (θ)

= tan^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= tan^{2} (θ)

= 1 + tan^{2} (θ)

Use the Pythagorean identity:

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

= sec^{2} (θ)

= sec^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

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