Answer:
77. [tex]\cot^{6} x = \cot^{4} x \csc^{2}x - \cot^{4} x[/tex]Proved
78. [tex]\sec^{4}x \tan^{2} x = \sec^{2}x [\tan^{2}x + \tan^{4}x ][/tex] Proved
79. [tex]\cos^{3} x\sin^{2} x = [\sin^{2}x - \sin^{4}x] \cos x[/tex] Proved.
80. [tex]\sin^{4}x - \cos^{4}x = 1 - 2\cos^{2}x + 2 \cos^{4} x[/tex] Proved.
Step-by-step explanation:
77. Left hand side
= [tex]\cot^{6} x[/tex]
= [tex]\cot^{4} x \times \cot^{2} x[/tex]
= [tex]\cot^{4}x [\csc^{2}x - 1][/tex]
{Since we know, [tex]\csc^{2} x - \cot^{2}x = 1[/tex]}
= [tex]\cot^{4} x \csc^{2}x - \cot^{4} x[/tex]
= Right hand side (Proved)
78. Left hand side
= [tex]\sec^{4}x \tan^{2} x[/tex]
= [tex]\sec^{2} x [1 + \tan^{2}x] \tan^{2} x[/tex]
{Since [tex]\sec^{2}x - \tan^{2}x = 1[/tex]}
= [tex]\sec^{2}x [\tan^{2}x + \tan^{4}x ][/tex]
= Right hand side (Proved)
79. Left hand side
= [tex]\cos^{3} x\sin^{2} x[/tex]
= [tex]\cos x[1 - \sin^{2} x] \sin^{2} x[/tex]
{Since [tex]\sin^{2}x + \cos^{2} x = 1[/tex]}
= [tex][\sin^{2}x - \sin^{4}x] \cos x[/tex]
= Right hand side
80. Left hand side
= [tex]\sin^{4}x - \cos^{4}x[/tex]
= [tex][\sin^{2}x + \cos^{2}x]^{2} - 2\sin^{2} x \cos^{2}x[/tex]
{Since [tex]\sin^{2}x + \cos^{2} x = 1[/tex]}
= [tex]1 - 2\cos^{2} x[1 - \cos^{2}x ][/tex]
= [tex]1 - 2\cos^{2}x + 2 \cos^{4} x[/tex]
= Right hand side. (Proved)
