Is tan(x)-tan(y)=(sin(x-y))/(cos(x)cos(y)) ?

The answer to whether tan(x)-tan(y)=(sin(x-y))/(cos(x)cos(y)) is True

Popular >

prove tan(x)-tan(y)=(sin(x-y))/(cos(x)cos(y))

prove

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

True

Solution steps

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

Manipulating left side

tan (x) - tan (y)

Express with sin, cos

= \frac{- cos ( x ) sin ( y ) + cos ( y ) sin ( x )}{cos ( x ) cos ( y )}

Manipulating right side

\frac{sin ( x - y )}{cos ( x ) cos ( y )}

Rewrite using trig identities

= \frac{sin ( x ) cos ( y ) - cos ( x ) sin ( y )}{cos ( x ) cos ( y )}

We showed that the two sides could take the same form

\Rightarrow True

sec^{2} (x) tan^{2} (x) + sec^{2} (x) = sec^{4} (x)

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

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

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

4 sin^{2} (θ) + 2 cos^{2} (θ) = 4 - 2 cos^{2} (θ)

Is tan(x)-tan(y)=(sin(x-y))/(cos(x)cos(y)) ?
The answer to whether tan(x)-tan(y)=(sin(x-y))/(cos(x)cos(y)) is True