Verify the following identify tan6 x6 c1 sec6 x6 sec4 x2

Verify the following identify: tan^6 x/6 + c_1 = sec^6 x/6 - sec^4 x/2 + sec^2 x/2 + c_2

Solution

tan⁶x/6 + c₁= sec⁶x/6 - sec⁴x/2 + sec²x/2 + c₂

LHS = tan⁶x/6 +   c₁ [tan²x = sec²x - 1]

we get

LHS = (sec²x - 1)³/6 + c₁ = sec⁶x/6 - 3sec⁴x/6 + 3sec²x/6 -1 + c₁

= sec⁶x/6 - sec⁴x/2 + sec²x/2 + c₁ - 1

= RHS for c₂ = c₁ -1