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# Find the value of tan^2 pi/16 + tan^2 3pi/16 + tan^2 5pi/16 +tan^2 7pi/16

Nishant Vora IIT Patna
5 years ago

Jainam Ravani
13 Points
one year ago
First convert the tan terms to cot using allied angles. You will get:
tan^2(π/16)+cot^2(π/16)+tan^2(3π/16)+cot^2(3π/16)
Taking π/16 as a, and converting to sin and cos, you get:
(Sin^4a+cos^4a)/sin^2acos^2a+(sin^4(3a)+cos^4(3a))/sin^2(3a)cos^2(3a)
=(1-sin^2acos^2a)/sin^2acos^2a+(1-sin^2(3a)cos^2(3a)/sin^2(3a)cos^2(3a)
Using the identity sin2a=2sinacosa:
=(4-2sin^2(2a))/sin^2(2a)+(4-2sin^2(6a))/sin^2(6a)
=4/sin^2(2a)+4/sin^2(2a)-4
=4(sin^(6a)+sin^(2a))/sin^2(2a)cos^2(2a)-4
sin^2(6a)=cos^2(2a). Using this and simplifying:
4/sin^2(2a)cos^2(2a)-4
Using the sin2a identity:
16/sin^2(4a)-4
=32-4
=28

Anurag Verma
13 Points
one year ago
tan^2(π/16)+tan^2(7π/16)=tan^(π/16)+cot^2(π/16)=(tana-cota)^2+2tana.cota=4cot^2a+2.similarly from 2nd and 6th term after simplifying =4cot^(π/4)+2 and from 3rd and 5th term 4cot^3π/8+2 and the remaining termis cot^45=1.  after that the simplifying lst and 3rd term will be simplified to 16cot^45+4. As a result the answer will be 16+8+4+6+1=35