$$\begin{gathered} {\text{For z = x + i y, we have}} \hfill \\\ {\text{sin z = ____________}}{\text{.}} \hfill \\\\ \end{gathered}$$

• A) $$\sin x\cosh y + i\cos x\sinh y$$
• B) $$\cos x\sinh y + i\sin x\cosh y$$
• C) $$\sin x\cos y + i\cos x\sin y$$
• D) $$\cos x\sin y + i\sin x\cos y$$

data:image/png;base64,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 .

• A) data:image/png;base64,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 .
• B) data:image/png;base64,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 .
• C)
• D)

We take the derivative of principal branch of zc because its principal branch is ______________ function.

• A) multivalued
• B) single valued
• C)
• D)

$${\text{For any complex number z, }}\frac{d}{{dz}}{\text{(sec z) = __________}}{\text{.}}$$

• A) $$- \sec z\tan z$$
• B) $$\tan z\sec z$$
• C) $$\cot z\sec z$$
• D) $$-\cot z\sec z$$

$${\text{For any complex number z, }}\frac{d}{{dz}}{\text{(tan z) = ________}}{\text{.}}$$

• A) $${\sec ^2}z$$
• B) $$- \sec z\tan z$$
• C) $$\tan z\sec z$$
• D) $$- {\csc ^2}z$$

$${\text{For any complex number z, }}\frac{d}{{dz}}{\text{csch }}z{\text{ = ___________}}{\text{.}}$$

• A) $$\sec {h^2}z$$
• B) $$- \csc {h^2}z$$
• C) $$- \csc hz\coth z$$
• D) $$- \sec hz\tanh z$$

$${\text{For any complex number z, }}\frac{d}{{dz}}{\text{(cot z) = _________}}{\text{.}}$$

• A) $$- {\csc ^2}z$$
• B) $$\tan z\sec z$$
• C) $$- \sec z\tan z$$
• D) $$- {\sec ^2}z$$

data:image/png;base64,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 .

• A) single valued
• B) multivalued
• C)
• D)

$$\begin{gathered} {\text{For z = x + i y, we have}} \hfill \ {\text{cos z = ___________}}{\text{.}} \hfill \\\ \end{gathered}$$

• A) $$\cos x\cosh y + i\sin x\sinh y$$
• B) $$\sin x\cosh y + i\cos x\sinh y$$
• C) $$\cos x\cosh y - i\sin x\sinh y$$
• D) $$\sin x\sinh y + i\cos x\cosh y$$

$${\text{For any complex number z, }}\frac{d}{{dz}}{\text{tanh }}z{\text{ = ____________}}{\text{.}}$$

• A) $$\sin z$$
• B) $$sechz\tanh z$$
• C) $$\sec {h^2}z$$
• D) $$\sinh z$$

$${\text{For any complex number z, }}\frac{d}{{dz}}{\text{cosh }}z{\text{ = ____________}}{\text{.}}$$

• A) $$\sin z$$
• B) $$\sec {h^2}z$$
• C) $$\sinh z$$
• D) $$\tanh z$$

$${\text{For any complex number z, sech }}z{\text{ = ___________}}{\text{.}}$$

• A) $$\frac{{{e^z} - {e^{ - z}}}}{2}$$
• B) $$\frac{2}{{{e^z} + {e^{ - z}}}}$$
• C) $$\frac{{{e^z} + {e^{ - z}}}}{2}$$
• D) $$\frac{2}{{{e^z} - {e^{ - z}}}}$$

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• A) finite values
• B) infinite values
• C)
• D)

$${\text{For any complex number z, }}\frac{d}{{dz}}{\text{coth }}z{\text{ = ____________}}{\text{.}}$$

• A) $$\sec {h^2}z$$
• B) $$- \csc {h^2}z$$
• C) $$- \csc hz\coth z$$
• D) $$\sinh z$$

$${\text{For any complex number z, tanh }}z{\text{ = ____________}}{\text{.}}$$

• A) $$\frac{{{e^{iz}} + {e^{ - iz}}}}{{{e^{iz}} - {e^{ - iz}}}}$$
• B) $$\frac{{{e^z} - {e^{ - z}}}}{{{e^z} + {e^{ - z}}}}$$
• C) $$\frac{{{e^z} + {e^{ - z}}}}{{{e^z} - {e^{ - z}}}}$$
• D) $$\frac{{{e^{iz}} - {e^{ - iz}}}}{{{e^{iz}} + {e^{ - iz}}}}$$

$${\text{For any complex number z, coth }}z{\text{ = _____________}}{\text{.}}$$

• A) $$\frac{{{e^z} - {e^{ - z}}}}{{{e^z} + {e^{ - z}}}}$$
• B) $$\frac{{{e^{iz}} - {e^{ - iz}}}}{{{e^{iz}} + {e^{ - iz}}}}$$
• C) $$\frac{{{e^{iz}} + {e^{ - iz}}}}{{{e^{iz}} - {e^{ - iz}}}}$$
• D) $$\frac{{{e^z} + {e^{ - z}}}}{{{e^z} - {e^{ - z}}}}$$

data:image/png;base64,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 .

• A) data:image/png;base64,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 .
• B) data:image/png;base64,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 .
• C) data:image/png;base64,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 .
• D) data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEkAAAAiCAYAAAATQPRFAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAHdElNRQfnBQsJDSReCCj4AAAAB3RFWHRBdXRob3IAqa7MSAAAAAx0RVh0RGVzY3JpcHRpb24AEwkhIwAAAAp0RVh0Q29weXJpZ2h0AKwPzDoAAAAOdEVYdENyZWF0aW9uIHRpbWUANfcPCQAAAAl0RVh0U29mdHdhcmUAXXD/OgAAAAt0RVh0RGlzY2xhaW1lcgC3wLSPAAAACHRFWHRXYXJuaW5nAMAb5ocAAAAHdEVYdFNvdXJjZQD1/4PrAAAACHRFWHRDb21tZW50APbMlr8AAAAGdEVYdFRpdGxlAKju0icAAAMjSURBVGhD7ZlPbuowEMa/vDuQRTesUbrolgMQvXNUcIGw6SWaCwRxB3YoHIBtF0Ssu2ERDpE3/hPixInj0EjQJ/8kqwke7PHnsWdQvYKAw8gf+ddhwIlkgRPJAieSBU4kC5xIFvwCka44rD14m0y+/5xsQ+OtDzSyJaxOelrytIgQFWku3wdwSsDqP6U1xhkwtiZSnkZy0KQ4yc8ew6lIyI9EcyIv0kguXO+UMJvSf2HfasqF6l9nayQJoR4rEvfBKAKKqDcMuoSu4BFnMiDaj9spebBIfYtj/X1HpV8gjsVan/Pizo5YIcLUl+9NeP8MLxP5rpFh420xzQssA7r6D2t03vvBHAmNdjTkhUEisck8jzJD2bSZZSZSbWQbnpy6RbhezrT5cwRcDDnHbQLmwystO0boiz4/nGFOYpk4X7pznbVITCA/BCjE2REFXRqIVq9YH6rBs42PECnoFHAbOu9EAgplvqO2cBEM5N8xxdkRa++IOZuLTbTaQrgyweJT+nhrSxK0Cx/TSD52YClShl0Y03rfsSh3d7LAO/kWhzvqZWQ4rijR/n0jNwXBnKl0hmGT7kDME5+n+CgX709JtBjfOTcYHTuRrhdaKgnQuCR8vgVji0D6v8zkUwvclwjpx+K2GRRaJJHhDjOSgwLTyIgXdwAWOFVk0Z6L0MJb5wVrol3869cecWNMcTxNF7kBGQAz05fpzOpoaVGk02Y9Iapa1U4p9Hi7t4zoTt96XdPumzUWJUCrSKKYrNchQhDlM16tqgUdE8iuzK+qeiEEG7tZGOpiMOQmRCk9CfSNGkb7PHV0kbiyYgHNyYVDVasvrBlFVVN94AKVi2RCk7BRq5Mt0cTt0+KkiKwKNpif/Cy5D7YoPZLqP3EaNn1O8n676BzMgLFHE6kWISrqmW+Iwr/TE+pkJSK0184efiIGROBo2Y2n7XiPr1pGoup3SxmOV8cNrgfs9uVzhqyzjJDF4ZBqtIdgSeN9KiVEH1KsUVAv5FurXyrVvcV2kkeWfJYWz4j756QFIxaT/y9OJAucSBY4kSxwIlngRLLAiWSBE6kX4B9TTxEPD+0KTwAAAABJRU5ErkJggg== .

$${\text{For any complex number z, }}\frac{d}{{dz}}{\text{sech }}z{\text{ = ___________}}{\text{.}}$$

• A) $$- \csc {h^2}z$$
• B) $$\sinh z$$
• C) $$- \sec hz\tanh z$$
• D) $$\sec {h^2}z$$

$${\text{For any complex number z, cosh }}z{\text{ = _____________}}{\text{.}}$$

• A) $$\frac{{{e^{iz}} + {e^{ - iz}}}}{2}$$
• B) $$\frac{{{e^z} + {e^{ - z}}}}{2}$$
• C) $$\frac{{{e^z} - {e^{ - z}}}}{2}$$
• D) $$\frac{{{e^{iz}} - {e^{ - iz}}}}{{2i}}$$

data:image/png;base64,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 .

• A) multivalued
• B) single valued
• C)
• D)