Prove that: (sin A + cosec A)^2 + (cos A+ sec A)^2 = 7+tan^2 A+ cotA^2
Draw a circle of radius 3 cm. Take two points P and Q on one of its extended
Answers
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Answer:
1)Prove that: (sin A + cosec A)^2 + (cos A+ sec A)^2 = 7+tan^2 A+ cotA^2
(sinA+cscA)
2
+(cosA+secA)
2
=sin
2
A+csc
2
A+2sinAcscA+cos
2
A+sec
2
A+2cosAsecA .......As[a²+b²+2ab=(a+b)²]
=sin
2
A+csc
2
A+2sinA×
sinA
1
+cos
2
A+sec
2
A+2cosA
cosA
1
.
........... since secA=
cosA
1
and cscA=
sinA
1
=sin
2
A+csc
2
A+2+cos
2
A+sec
2
A+2
=(sin
2
A+cos
2
A)+csc
2
A+sec
2
A+4
=1+1+cot
2
A+1+tan
2
A+4 ........... since csc
2
A=1+cot
2
A and sec
2
A=1+tan
2
A
=7+tan
2
A+cot
2
A
Hence proved.
2) Draw a circle of radius 3 cm. Take two points P and Q on one of its extended
Step 1: Draw a circle with centre O and radius 3 cm using a compass.
Step 2: Draw a secant passing through the centre. Mark points P and Q on opposite sides of the centre at a distance of 7 cm from O.
Step 3: Place the compass on P and draw two arcs on opposite sides of OP. Now place the compass on O and draw two arcs intersecting the arcs drawn from point P.
Step 4: Join the intersection points of the arcs to obtain the perpendicular bisector of OP. Mark the mid point of OP as M
1
Step 5: From M
1
draw a circle with radius =M
1
P=M
1
O
Step 6: Mark the intersection points of the circle drawn from M
1
with the circle drawn from O as A and B.
Step 7: Join P−A and P−B
Step 8: Repeat steps 3 to 7 for point Q and obtain tangents QC and QD
PA,PB,QC and QD are the required tangents.
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