Question

If sec theta - cos theta =2, then find sec^4 theta + cos^4 theta

Answer 1

The value of $\sec^4 \theta +\cos^4 \theta=34$.

Step-by-step explanation:

We have,

$\sec \theta-\cos \theta=2$                           ......(1)

To find, the value of $\sec^4 \theta +\cos^4 \theta=?$

Squaring (1) in both sides, we get

$(\sec \theta-\cos \theta)^{2} =2^{2}$

⇒$\sec ^{2} \theta+\cos \theta^{2} -2\sec\theta\cos \theta=4$

⇒$\sec ^{2} \theta+\cos \theta^{2} -2(1)=4$

[ ∵$sec\theta\cos \theta=1$]

⇒$\sec ^{2} \theta+\cos ^{2} \theta=4+2=6$    ....(2)

Again, squaring (2) in both sides, we get

$(\sec ^{2} \theta+\cos^{2} \theta )^{2} =6^{2}$

⇒ $\sec ^{4} \theta+\cos \theta^{4} +2\sec^{2} \theta\cos^{2} \theta=36$

⇒$\sec ^{4} \theta+\cos \theta^{4} +2(1)=36$

⇒$\sec ^{4} \theta+\cos \theta^{4} =36-2=34$

Hence, the value of $\sec^4 \theta +\cos^4 \theta=34$.