Question

prove the identity. sec⁴x tan⁶x=(tan⁶x + tan⁸x)sec²x note that each statement must be based on a rule chosen from the rule menu. to see a detailed description of the right of the rule. statement reason = tan⁶x + 2tan⁸x + tan¹⁰x algebra = tan⁶x(1 + 2tan²x + tan⁴x) algebra = select rule validate

Explanation:

Step1: Start with the right - hand side

$(\tan^{6}x+\tan^{8}x)\sec^{2}x$

Step2: Recall the identity $\sec^{2}x = 1+\tan^{2}x$

$(\tan^{6}x+\tan^{8}x)(1 + \tan^{2}x)$

Step3: Expand the product using distributive property (algebra)

$\tan^{6}x\times1+\tan^{6}x\times\tan^{2}x+\tan^{8}x\times1+\tan^{8}x\times\tan^{2}x$

Step4: Simplify using the rule of exponents $a^{m}\times a^{n}=a^{m + n}$

$\tan^{6}x+\tan^{8}x+\tan^{8}x+\tan^{10}x=\tan^{6}x + 2\tan^{8}x+\tan^{10}x$

Step5: Factor out $\tan^{6}x$

$\tan^{6}x(1 + 2\tan^{2}x+\tan^{4}x)$

Step6: Recall the perfect - square formula $(a + b)^{2}=a^{2}+2ab + b^{2}$, here $a = 1$ and $b=\tan^{2}x$

$\tan^{6}x(1+\tan^{2}x)^{2}$

Step7: Since $\sec^{2}x=1 + \tan^{2}x$

$\tan^{6}x(\sec^{2}x)^{2}=\sec^{4}x\tan^{6}x$

Answer:

The identity $\sec^{4}x\tan^{6}x=(\tan^{6}x+\tan^{8}x)\sec^{2}x$ is proved.