3. Prove that: = 1 cos (27t + O) co...
Question

# 3. Prove that: = 1 cos (27t + O) cosec (211 + ) tan (nt/2+0) sec (a/2+) cos O cot (T1 + ) cosec (90° + 0) + cot (450° + 0) tan (180° + 2) + sec (180° - 0) cosec (90° - 0) + tan (180° - 0) tan (360° + 0) - sec (0)

JEE/Engineering Exams
Maths
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( frac{cos (2 pi+theta) cos e c(2 pi+theta) tan left(frac{pi}{2}+thetaright)}{sec left(frac{pi}{2}+thetaright) cos theta cot (pi+theta)} ) ( =frac{(cos (2 pi+theta))left(frac{1}{sin (2 pi+theta)}right)left(frac{sin left(frac{pi}{2}+thetaright)}{cos left(frac{pi}{2}+thetaright)}right)}{left(frac{1}{cos left(frac{pi}{2}+thetaright)}right)(cos theta)left(frac{cos (pi+theta)}{sin (pi+theta)}right)} ) ( =frac{(cos (2 pi+theta))left(frac{1}{sin (2 pi+theta)}right) sin left(frac{pi}{2}+thetaright)}{(cos theta)left(frac{cos (pi+theta)}{sin (pi+theta)}right)} ) ( =frac{cos (pi+(pi+theta))left(frac{1}{sin (pi+(pi+theta))}right) sin left(frac{pi}{2}+thetaright)}{(cos theta)left(frac{cos (pi+theta)}{sin (pi+theta)}right)} ) ( =frac{-cos (pi+theta)left(-frac{1}{sin (pi+theta)}right) cos theta}{(cos theta)left(frac{-cos theta}{-sin theta}right)} quad(text {Since}, sin (pi+theta)=-sin theta text { and } cos (pi+theta)=-cos theta) ) ( =frac{cos thetaleft(frac{1}{sin theta)}right) cos theta}{(cos theta)left(frac{cos theta}{sin theta}right)}=1 ) Hence,( frac{cos (2 pi+theta) operatorname{cosec}(2 pi+theta) tan left(frac{pi}{2}+thetaright)}{sec left(frac{pi}{2}+thetaright) cos theta cot (pi+theta)}=1 )