If #7\sin ^ { 2} \theta + 3\cos ^ { 2} \theta = 4#, what is the value of #tan \theta#?

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Answer 1 · 2017-08-10T16:24:09

#7\sin ^ { 2} \theta + 3\cos ^ { 2} \theta = 4#

#=>7\sin ^ { 2} \theta + 3\cos ^ { 2} \theta = 4(sin^2theta+cos^2theta)#

#=>7\sin ^ { 2} \theta + 3\cos ^ { 2} \theta = 4sin^2theta+4cos^2theta#

#=>7\sin ^ { 2} \theta-4sin^2theta = 4cos^2theta- 3\cos ^ { 2} \theta#

#=>3sin^2theta = cos^2theta#

#=>tan^2theta=1/3#

#=>tantheta=pm1/sqrt3#

Alternative Way

Dividing both sides by #cos^2theta#

#7sin^2theta/cos^2theta + 3cos ^ 2theta/cos^2theta = 4/cos^2theta#

#=>7tan^2theta + 3 = 4sec^2theta#

#=>7tan^2theta + 3 = 4(1+tan^2theta)#

#=>7tan^2theta -4tan^2theta= 4-3=1#

#=>3tan^2theta=1#

#=>tantheta=pm1/sqrt3#