What killed Takeda Shingen?
A sniper's bullet?
A combination of the above? Why is the cause of his death shrouded in mystery? Did the Takeda try and keep his death secret or conceal the cause?
It's most likely illness. Takeda Shingen's death has been reported as pulmonary tuberculosis in the Buke Jiki,
武家事紀, and as throat or stomach cancer in the Kōyō Gunkan
甲陽軍鑑. There's a theory that he was killed by a bullet or bullet wound, but this is generally regarded as a popular myth. It is only found in pro-Tokugawa writings (i.e. propaganda) produced during the Edo period, where the ruling Tokugawa clan would have liked to be credited with indirectly killing the great Shingen.
Shingen did actually leave orders to keep his death secret for three years. I don't think they fooled anyone, though.