\( (\sec A-\cos A)(\cot A+\tan A)=\sec A \tan A \)

1.	\( (\sec A-\cos A)(\cot A+\tan A)=\sec A \tan A \)
Answer» (sec A − cos A) (cot A + tan A) = sec A . cot A + sec A . tan A - cos A.cot A - cosA . tan A = \(\frac{1}{cos\,A}\) . \(\frac{cos\,A}{sin\,A}\) + sec A . tan A - cos A . \(\frac{cos\, A}{sin\,A}-cos\,A.\frac{sin\,A}{cos\,A}\) = \(\frac{1}{sin\,A}-sin\,A-\frac{cos^2A}{sin\,A}+sec\,A.tan\,A\) = \(\frac{1-sin^2\,A-cos^2\,A}{sin\,A}\) + sec A . tan A = \(\frac{1-1}{sin\,A}\) + sec A . tan A (∵ sin²A + cos²A = 1) = sec A . tan A

Answer»

(sec A − cos A) (cot A + tan A)

= sec A . cot A + sec A . tan A - cos A.cot A - cosA . tan A

= \(\frac{1}{cos\,A}\) . \(\frac{cos\,A}{sin\,A}\) + sec A . tan A - cos A . \(\frac{cos\, A}{sin\,A}-cos\,A.\frac{sin\,A}{cos\,A}\)

= \(\frac{1}{sin\,A}-sin\,A-\frac{cos^2A}{sin\,A}+sec\,A.tan\,A\)

= \(\frac{1-sin^2\,A-cos^2\,A}{sin\,A}\) + sec A . tan A

= \(\frac{1-1}{sin\,A}\) + sec A . tan A

(∵ sin²A + cos²A = 1)

= sec A . tan A

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\( (\sec A-\cos A)(\cot A+\tan A)=\sec A \tan A \)

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