| Assertion (A): \[({{\cos }^{2}}\theta -{{\sin }^{2}}\theta )=\frac{2\tan \theta }{(1-{{\tan }^{2}}\theta )}\] is not an identity. ; |
| Reason (R): A equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of angles involved. |
A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A)
B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A)
C) Assertion (A) is true but Reason (R) is false
D) Assertion (A) is false but Reason (R) is true
Correct Answer: A
Solution :
| [a] We have , \[(co{{s}^{2}}\theta -si{{n}^{2}}\theta )=\ \ \frac{2\tan \theta }{1-{{\tan }^{2}}\theta }\] |
| Putting \[\theta ={{30}^{o}},\operatorname{we}\ get\ L.H.S=(co{{s}^{2}}{{30}^{o}}-si{{n}^{2}}{{30}^{o}})\] |
| \[\left\{ {{\left( \frac{\sqrt{3}}{2} \right)}^{2}}-{{\left( \frac{1}{2} \right)}^{2}} \right\}=\left( \frac{3}{4}-\frac{1}{4} \right)=\frac{2}{4}=\frac{1}{2}\] |
| \[\operatorname{R}.H.S=\frac{2\tan {{30}^{o}}}{(1-ta{{n}^{2}}{{30}^{o}})}=\frac{2\times \frac{1}{\sqrt{3}}}{\left( 1-\frac{1}{3} \right)}=\left( \frac{2}{\sqrt{3}}\times \frac{3}{2} \right)=\sqrt{3}\] |
| \[\therefore \operatorname{L}.H.S\ne \operatorname{R}.H.S\] |
| Hence the given equation is not an identity. |
| \[\therefore \] Assertion true: Reason: True and it is the correct explanation of Assertion. |
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