Question

Please help me with these, oh sweet jesus

Answer 1

77.
`\cot^(6) x = \cot^(4) x \csc^(2)x - \cot^(4) x`Proved

78.
`\sec^(4)x \tan^(2) x = \sec^(2)x [\tan^(2)x + \tan^(4)x ]` Proved

79.
`\cos^(3) x\sin^(2) x = [\sin^(2)x - \sin^(4)x] \cos x` Proved.

80.
`\sin^(4)x - \cos^(4)x = 1 - 2\cos^(2)x + 2 \cos^(4) x` Proved.

Explanation:

77. Left hand side

=
`\cot^(6) x`

=
`\cot^(4) x * \cot^(2) x`

=
`\cot^(4)x [\csc^(2)x - 1]`

{Since we know,
`\csc^(2) x - \cot^(2)x = 1`}

=
`\cot^(4) x \csc^(2)x - \cot^(4) x`

= Right hand side (Proved)

78. Left hand side

=
`\sec^(4)x \tan^(2) x`

=
`\sec^(2) x [1 + \tan^(2)x] \tan^(2) x`

{Since
`\sec^(2)x - \tan^(2)x = 1`}

=
`\sec^(2)x [\tan^(2)x + \tan^(4)x ]`

= Right hand side (Proved)

79. Left hand side

=
`\cos^(3) x\sin^(2) x`

=
`\cos x[1 - \sin^(2) x] \sin^(2) x`

{Since
`\sin^(2)x + \cos^(2) x = 1`}

=
`[\sin^(2)x - \sin^(4)x] \cos x`

= Right hand side

80. Left hand side

=
`\sin^(4)x - \cos^(4)x`

=
`[\sin^(2)x + \cos^(2)x]^(2) - 2\sin^(2) x \cos^(2)x`

{Since
`\sin^(2)x + \cos^(2) x = 1`}

=
`1 - 2\cos^(2) x[1 - \cos^(2)x ]`

=
`1 - 2\cos^(2)x + 2 \cos^(4) x`

= Right hand side. (Proved)