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Happy New Year to you and wishing you much success in 2017. Let's test some q-function identities mentioned in István Mezö, "Several special values of Jacobi theta functions" https://arxiv.org/pdf/1106.2703.pdf .
C:\Temp>qconsole
>(SetPrecision 150)
150
>(Setq q (Random))
.355137292830204446040446680586593508063560573
>(Setq z (Random))
.203417555285391595827949454245639793772563901
>(QCos q z)
.979381891475308569875632622475047204341036211
>(QSin q (- (/ _Pi 2) z))
.979381891475308569875632622475047204341036211
>(QSin q (* 2 z))
.395704743499764397816126636850458208601633875
>(* (^ q -1/4)
(^ (QPochhammer (^ q 2) (^ q 4) _Infinity) 4)
(^ (QPochhammer q (^ q 2) _Infinity) -2)
(QSin (^ q 2) z)
(QCos (^ q 2) z))
.395704743499764397816126636850458208601633876
>(QSin q (* 2 z))
.395704743499764397816126636850458208601633875
>(* (^ q -1/4)
(/ (^ (QEuler (^ q 2)) 6)
(* (^ (QEuler q) 2) (^ (QEuler (^ q 4)) 4)))
(QSin (^ q 2) z)
(QCos (^ q 2) z))
.395704743499764397816126636850458208601633875
>(QCos q (* 2 z))
.918377778967351404206934515211872228766801206
>(- (^ (QCos (^ q 2) z) 2) (^ (QSin (^ q 2) z) 2))
.918377778967351404206934515211872228766801206
>(QSin q z)
.202017591730828019212444730769624786518875034
>(* (^ q 1/4)
(^ (QGamma (^ q 2) 1/2) 2)
(/ (^ (^ q 2) (Binomial (/ z _Pi) 2))
(* (QGamma (^ q 2) (/ z _Pi))
(QGamma (^ q 2) (- 1 (/ z _Pi))))))
.202017591730828019212444730769624786518875042
>(QSin (^ q 2) (/ _Pi 4))
.707004423594506121995272278335598201755934937
>(Def (C q)
(* (^ q -1/4)
(/ (^ (QEuler (^ q 2)) 6)
(* (^ (QEuler q) 2) (^ (QEuler (^ q 4)) 4))))
)
C
>(^ (C q) -1/2)
.707004423594506121995272278335598201755934937
>(QSin (^ q 2) (/ _Pi 8))
.382588866281506265790050905283644236435230184
>(Sqrt (* (QSin q (/ _Pi 4))
(/ (- (Sqrt (+ 1 (/ 4 (^ (C q) 2)))) 1)
2)))
.382588866281506265790050905283644236435230184
>
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Jan 1, 2017 - New Year's Day
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Hello, PLANETQUANTUM.COM visitor.