Is tan(pi/2-x)sin(x)=cos(x) ?

The answer to whether tan(pi/2-x)sin(x)=cos(x) is True

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prove tan(pi/2-x)sin(x)=cos(x)

prove

tan (\frac{π}{2} - x) sin (x) = cos (x)

True

Solution steps

tan (\frac{π}{2} - x) sin (x) = cos (x)

Manipulating left side

tan (\frac{π}{2} - x) sin (x)

Rewrite using trig identities

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

Simplify

\frac{cos ( x )}{sin ( x )} sin (x) : cos (x)

= cos (x)

We showed that the two sides could take the same form

\Rightarrow True

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

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

\frac{1 + cos ( θ )}{1 + sec ( θ )} = cos (θ)

tan^{2} (θ) - sec^{2} (θ) = - 1

cot (x) sin (x) - cos^{2} (x) sec (x) = 0

Is tan(pi/2-x)sin(x)=cos(x) ?
The answer to whether tan(pi/2-x)sin(x)=cos(x) is True