SOLUTION: Find the exact value of the expression: {{{ sin ( pi/6 ) + tan^2 ( pi/5 ). }}} I am familiar with some of the trigonometric identities, however for some reason this one has me s

Algebra ->  Trigonometry-basics -> SOLUTION: Find the exact value of the expression: {{{ sin ( pi/6 ) + tan^2 ( pi/5 ). }}} I am familiar with some of the trigonometric identities, however for some reason this one has me s      Log On


   

Question 962295: Find the exact value of the expression: +sin+%28+pi%2F6+%29+%2B+tan%5E2+%28+pi%2F5+%29.%0D%0A
I am familiar with some of the trigonometric identities, however for some reason this one has me stumped. I attempted to convert them into their reciprocals, cosecant and cotangent in order to try and see if it lined up with the following identity.
csc^2(x) = cot^2(x) + 1
That did not seem to work. I think what's messing me up here is the tangent being squared and the sine is not. Unfortunately, with this expression, I'm unsure of where to start.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
I do not think you need to use identities to solve this
+sin+%28+pi%2F6+%29+%2B+tan%5E2+%28+pi%2F5+%29
since +sin+%28+pi%2F6+%29=1%2F2, and
tan%5E2+%28+pi%2F5+%29=tan+%28+pi%2F5+%29%2Atan%28+pi%2F5+%29, and
tan+%28+pi%2F5+%29=sqrt%285-2+sqrt%285%29%29 then
tan%5E2+%28+pi%2F5+%29=%28sqrt%285-2sqrt%285%29%29%29%5E2
tan%5E2+%28+pi%2F5+%29=5-2sqrt%285%29
so we have:
+1%2F2+%2B+%285-2sqrt%285%29%29
+1%2F2+%2B+10%2F2-2sqrt%285%29%29
+11%2F2-2sqrt%285%29%29
the exact value of the expression is:11%2F2-2sqrt%285%29