perform the indicated operation and simplify the result so that there are no quotients. \n\\( \\frac { 1 } {…

perform the indicated operation and simplify the result so that there are no quotients. \n\\( \\frac { 1 } { \\csc ^ { 2 } \\theta } + \\frac { 1 } { \\sec ^ { 2 } \\theta } \\)\n\\( \\frac { 1 } { \\csc ^ { 2 } \\theta } + \\frac { 1 } { \\sec ^ { 2 } \\theta } = \\square \\) (simplify your answer.)

perform the indicated operation and simplify the result so that there are no quotients. \n\\( \\frac { 1 } { \\csc ^ { 2 } \\theta } + \\frac { 1 } { \\sec ^ { 2 } \\theta } \\)\n\\( \\frac { 1 } { \\csc ^ { 2 } \\theta } + \\frac { 1 } { \\sec ^ { 2 } \\theta } = \\square \\) (simplify your answer.)

Answer

Explanation:

Step1: Use reciprocal identities

Recall that (\csc\theta=\frac{1}{\sin\theta}), so (\frac{1}{\csc^{2}\theta}=\sin^{2}\theta); and (\sec\theta = \frac{1}{\cos\theta}), so (\frac{1}{\sec^{2}\theta}=\cos^{2}\theta). $$\frac{1}{\csc^{2}\theta}+\frac{1}{\sec^{2}\theta}=\sin^{2}\theta+\cos^{2}\theta$$

Step2: Use Pythagorean identity

By the Pythagorean identity (\sin^{2}\theta+\cos^{2}\theta = 1)

Answer:

(1)