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Question
State whether the statement is True or False? Also give justification.
If tan(π cosθ) = cot(π sinθ), then `cos(theta - pi/4) = +- 1/(2sqrt(2))`.
Options
True
False
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Solution
This statement is True.
Explanation:
Given that: tan(π cosθ) = cot(π sinθ)
⇒ tan(π cosθ) = `tan(pi/2 - pi sin theta)`
⇒ πcosθ = `pi/2 - pi sin theta`
⇒ πcosθ + πsinθ = `pi/2`
⇒ cosθ + sinθ = `1/2`
⇒ `1/sqrt(2) cos theta + 1/sqrt(2) sin theta = 1/(2sqrt(2))`
⇒ `cos pi/4 cos theta + sin pi/4 sin theta = 1/(2sqrt(2))`
⇒ `cos(theta - pi/4) = +- 1/(2sqrt(2))` ......`[because cos(theta - pi/2) "or" cos(pi/4 - theta)]`
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