Question 142

If $$\ \frac{1}{cos\theta+sec\theta}=\frac{1}{2}\ $$, then what is the value of $$\ cos^{100}\ \theta+sec^{100}\ \theta\ $$?

Solution

Given : $$\ \frac{1}{cos\theta+sec\theta}=\frac{1}{2}\ $$

=> $$\ \frac{1}{cos\theta+\frac{1}{cos\theta}}=\frac{1}{2}\ $$

=> $$\ \frac{cos\theta}{cos^2\theta+1}=\frac{1}{2}\ $$

=> $$cos^2\theta+1-2cos\theta=0$$

=> $$(cos\theta-1)^2=0$$

=> $$cos\theta=1$$

Also, $$sec\theta=\frac{1}{cos\theta}=1$$

$$\therefore$$ $$\ cos^{100}\ \theta+sec^{100}\ \theta\ $$

= $$(1)^{100}+(1)^{100}=1+1=2$$

=> Ans - (C)

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