Question

Solve the above que no. 55

Answer 1

Let
`\left(1+(1)/(\tan^(2)A) \right)\cdot \left(1+(1)/(\cot^(2)A) \right)`, we proceed to prove the trigonometric expression by trigonometric identity:

1)
`\left(1+(1)/(\tan^(2)A) \right)\cdot \left(1+(1)/(\cot^(2)A) \right)` Given

2)
`\left(1+(\cos^(2)A)/(\sin^(2)A) \right)\cdot \left(1+(\sin^(2)A)/(\cos^(2)A) \right)`
`\tan A = (1)/(\cot A) = (\sin A)/(\cos A)`

3)
`\left((\sin^(2)A+\cos^(2)A)/(\sin^(2)A) \right)\cdot \left((\cos^(2)A+\sin^(2)A)/(\cos^(2)A) \right)`

4)
`\left((1)/(\sin^(2)A) \right)\cdot \left((1)/(\cos^(2)A) \right)`
`\sin^(2)A+\cos^(2)A = 1`

5)
`(1)/(\sin^(2)A\cdot \cos^(2)A)`

6)
`(1)/(\sin^(2)A\cdot (1-\sin^(2)A))`
`\sin^(2)A+\cos^(2)A = 1`

7)
`(1)/(\sin^(2)A-\sin^(4)A)` Result

Explanation:

Let
`\left(1+(1)/(\tan^(2)A) \right)\cdot \left(1+(1)/(\cot^(2)A) \right)`, we proceed to prove the trigonometric expression by trigonometric identity:

1)
`\left(1+(1)/(\tan^(2)A) \right)\cdot \left(1+(1)/(\cot^(2)A) \right)` Given

2)
`\left(1+(\cos^(2)A)/(\sin^(2)A) \right)\cdot \left(1+(\sin^(2)A)/(\cos^(2)A) \right)`
`\tan A = (1)/(\cot A) = (\sin A)/(\cos A)`

3)
`\left((\sin^(2)A+\cos^(2)A)/(\sin^(2)A) \right)\cdot \left((\cos^(2)A+\sin^(2)A)/(\cos^(2)A) \right)`

4)
`\left((1)/(\sin^(2)A) \right)\cdot \left((1)/(\cos^(2)A) \right)`
`\sin^(2)A+\cos^(2)A = 1`

5)
`(1)/(\sin^(2)A\cdot \cos^(2)A)`

6)
`(1)/(\sin^(2)A\cdot (1-\sin^(2)A))`
`\sin^(2)A+\cos^(2)A = 1`

7)
`(1)/(\sin^(2)A-\sin^(4)A)` Result