Solution
= Solution
Because $d(T_1)$ commutes with itself and is anti-Hermitian,
$$
\langle\varphi|\theta\rangle
=\langle\Lambda|e^{\varphi d(T_1)}e^{-\theta d(T_1)}|\Lambda\rangle
=\langle\Lambda|e^{-(\theta-\varphi)d(T_1)}|\Lambda\rangle.
$$
Applying the preceding highest-weight matrix coefficient yields
$$
\boxed{\langle\varphi|\theta\rangle
=\left[\cos\frac{\theta-\varphi}{2}\right]^\Lambda}.
$$