1.

What is the simplified value of \(\left[ {\frac{{\cos A}}{{(1 - \tan A)\;}} + {{\frac{{\sin A}}{{(1 - \cot A)\;}}}}} \right]^2?\)1). sin A + cos A2). 1 + sin 2A3). 1 + cos 2A4). tan A + cot A

Answer»

Put tan A = sin A/cos A and COT A = cos A/sin A 

⇒ $({\left( {\frac{{cosA}}{{1 - \frac{{SINA}}{{cosA}}}}\; + \;\frac{{sinA}}{{1 - \frac{{cosA}}{{sinA}}}}} \right)^2})$

⇒ $({\left( {\frac{{{{\cos }^2}A}}{{\cos A - \sin A}}\; - \;\frac{{{{\sin }^2}A}}{{\cos A - \sin A}}} \right)^2})$

⇒ $({\left( {\frac{{\left( {cosA - sinA} \right)\;\left( {\cos A\; + \;\sin A} \right)\;}}{{\cos A - \sin A}}} \right)^2})$

⇒ (cosA + sinA) 2

⇒ cos2A + sin2A + 2sinA cos A, Since we know that cos2A + cos2A = 1 and 2sinA cosA = SIN2A

⇒ 1 + sin2A

∴ its simplified value is 1 + sin2A



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