Question 94

What is the simplified value of $$sec^6 A - tan^6 A - 3 sec^2 A tan^2A$$?

Solution

Expression : $$sec^6 A - tan^6 A - 3 sec^2 A tan^2A$$

= $$[(sec^2A)^3-(tan^2A)^3]-3sec^2Atan^2A$$

Using, $$x^3-y^3=(x-y)(x^2+y^2+xy)$$

= $$[(sec^2A-tan^2A)(sec^4A+tan^4A+sec^2Atan^2A)]-3sec^2Atan^2A$$

Also, $$(sec^2A-tan^2A=1)$$

= $$sec^4A+tan^4A-2sec^2Atan^2A$$

Using, $$x^2+y^2-2xy=(x-y)^2$$

= $$(sec^2A-tan^2A)^2$$

= $$(1)^2=1$$

=> Ans - (C)

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