syms x
sin2exp = rewrite(sin(x), 'exp')
tan2exp = rewrite(tan(x), 'exp')

sin2exp =
(exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2

tan2exp =
-(exp(x*2i)*1i - 1i)/(exp(x*2i) + 1)

Rewrite the exponential function in terms of any trigonometric function by specifying the trigonometric function as the target. For a full list of targets, see target.

syms x
exp2sin = rewrite(exp(x), 'sin')
exp2tan = rewrite(-(exp(x*2i)*1i - 1i)/(exp(x*2i) + 1), 'tan')

exp2sin =
1 - 2*sin((x*1i)/2)^2 - sin(x*1i)*1i
exp2tan =
-(((tan(x) - 1i)*1i)/(tan(x) + 1i) + 1i)/...
    ((tan(x) - 1i)/(tan(x) + 1i) - 1)

Simplify exp2tan into the expected form by using simplify.

exp2tan = simplify(exp2tan)

exp2tan =
tan(x)

Rewrite tan(x) in terms of the sine function by specifying the target 'sin'.

syms x
tan2sin = rewrite(tan(x), 'sin')

tan2sin =
-sin(x)/(2*sin(x/2)^2 - 1)

Rewrite tanh(x) in terms of the sine function by specifying the target 'sin'.

syms x
tanh2sin = rewrite(tanh(x), 'sin')

tanh2sin =
(sin(x*1i)*1i)/(2*sin((x*1i)/2)^2 - 1)

Similarly, rewrite trigonometric functions in terms of hyperbolic functions by specifying the hyperbolic function as the target.

Rewrite acos(x) and acot(x) in terms of the log function.

syms x
acos2log = rewrite(acos(x), 'log')
acot2log = rewrite(acot(x), 'log')

acos2log =
-log(x + (1 - x^2)^(1/2)*1i)*1i

acot2log =
(log(1 - 1i/x)*1i)/2 - (log(1i/x + 1)*1i)/2

Similarly, rewrite the logarithm function in terms of an inverse trigonometric function by specifying the inverse trigonometric function as the target.

Rewrite all elements of a matrix in terms of the exp function.

syms x
matrix = [sin(x) cos(x); sinh(x) cosh(x)];
rewrite(matrix, 'exp')

ans =
[ (exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2, exp(-x*1i)/2 + exp(x*1i)/2]
[                 exp(x)/2 - exp(-x)/2,       exp(-x)/2 + exp(x)/2]

syms x
rewrite(cos(x),'sin')

ans =
1 - 2*sin(x/2)^2

rewrite does not replace sin(x) with either $$-\sqrt{1-{\cos }^2\left(x\right)}$$ or $$\sqrt{1-{\cos }^2\left(x\right)}$$ because these expressions are not valid for all x. However, using the square of these expressions to replace sin(x)^2 is valid for all x. Thus, rewrite replaces sin(x)^2.

syms x
rewrite(sin(x),'cos')
rewrite(sin(x)^2,'cos')

ans =
sin(x)
ans =
1 - cos(x)^2